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三个秘密

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Three Secrets

本书的目的是改变你看待世界的方式。

The aim of this book is to change the way you see the world.

它植根于我的个人旅程，一段漫长的冒险，它改变了我的身体，赋予了我神奇的力量。但这段旅程不属于我一个人。这是一段集体的旅程，是有史以来最古老、最强大的旅程之一。它由少数人从时间之初开始，一直持续到今天，改变着我们的文明、语言和思想。

It is grounded in my personal journey, a long adventure that has physically transformed me and endowed me with magical powers. But this journey is not mine alone. It is a collective journey, one of the most ancient and powerful ever. Begun at the dawn of time by a handful of human beings, it continues to this very day to transform our civilization, language, and thought.

我们当中有多少人感受到数学在我们心中生生不息、不断成长？我不知道。我只知道我们是极少数，我们的故事至今仍被误解。

How many among us have felt mathematics live and grow within ourselves? I don’t know. I just know that we are a tiny minority and that our story is as yet misunderstood.

数学素来以难以接近而闻名。只有精英才能获得特殊天赋。最伟大的数学家曾写道，事实并非如此。正如我们将看到的，他们声称自己所取得的成就是通过普通人类的手段、好奇心和想象力、怀疑和弱点取得的。

Mathematics has the reputation of being inaccessible. You have to be one of the elite, to have received a special gift. The greatest mathematicians have written that this isn’t so. What they accomplished, as we shall see, they claim to have accomplished through ordinary human means, their curiosity and imagination, their doubts and weaknesses.

没人愿意相信他们。也许他们不知道如何用足够简单的语言讲述他们的故事。或者也许他们低估了他们质疑的神话的力量，这是人类最后的伟大神话之一：智慧神话。

No one wanted to believe them. Perhaps they didn’t know how to tell their story in simple enough language. Or perhaps they underestimated the force of the myth that they called into question, one of the last great human myths: the myth of intelligence.

数学塑造了我们的世界。它是权力和统治的工具。但对于那些生活在数学中的人来说，数学首先是一种内心的体验，一种感官和精神的探索。

Mathematics shapes our world. It is an instrument of power and domination. But for those who live it, math is above all an inner experience, a sensual and spiritual quest.

这种体验与我们在学校学到的东西没什么关系。从某种程度上说，这是一种洞察力，一种心灵感应。从其他方面来说，这是一种神秘现象的复苏，这种现象让我们在童年时期学会了说话。

This experience has little to do with what we’re taught in school. In certain ways, it’s a form of clairvoyance, of psychic thought. In other ways, it’s a resurgence of the mysterious phenomenon that in our childhood allows us to learn how to speak.

理解数学就是沿着一条秘密的道路前行，这条道路会带我们回到孩提时代的心理可塑性。这条道路会带我们发现如何重新激活和驯化这种可塑性。这条道路会让我们选择让它重获生机。这条智力之路与我们日常生活中所走的道路惊人地相似。但它的入口是隐藏的，隐藏在我们的习惯、恐惧和压抑背后。我想帮助你找到这条路。

Understanding math is to travel along a secret path that brings us back to the mental plasticity we had as children. It’s to discover how to reactivate and domesticate that plasticity. It’s to choose to bring it back to life. This intellectual path is surprisingly close to that which we take in our everyday lives. But its entrance is hidden, concealed behind our habits, behind our fears and inhibitions. I would like to help you find this path.

一定还有其他原因

There Has to Be Something Else

“我没有什么特殊才能。我只是充满好奇心。”

“I have no special talent. I am only passionately curious.”

十五岁时，我很讨厌爱因斯坦的这句话。对我来说，这句话听起来很虚伪，很不真诚，就像一个超级名模说真正重要的是内在美。我们真的需要听这些吗？

When I was fifteen, I hated this quote from Einstein. To me it sounded phony, insincere, like a supermodel saying that what really counts is inner beauty. Do we really need to hear this stuff?

然而，这本书的主要信息是认真对待爱因斯坦的话。

The main message of this book, however, is to take Einstein’s words seriously.

仔细想想，我们竟然如此难以认真对待他，这真是令人惊讶。爱因斯坦并没有被冠以十足的白痴或惯性撒谎者的名声。如果你在街上问路人，他们会说他的相对论是对人类思想的伟大贡献之一。因此，爱因斯坦所说和所写的东西值得我们关注。

When you think about it, it is surprising that we have such a hard time taking him seriously. Einstein doesn’t have the reputation of being a complete idiot or a compulsive liar. If you ask people in the street, they’ll say that his theory of relativity is one of the great contributions to human thought. What Einstein said and wrote, therefore, merits our attention.

但当他暗示他的创造力可能对其他人有用时，我们很难相信他，因为他的创造力只是来自一种任何人都可以采取的略有不同的方法。这个可怜的老家伙不知道自己在说什么。或者更糟的是，这是虚伪的谦虚，他这么说只是为了炫耀。

But when he suggests that his creativity might be accessible to others, that it simply follows from a slightly different approach that anyone might take, we find it hard to believe him. The poor old guy doesn’t know what he’s saying. Or worse, it’s false modesty, and he’s saying it just to show off.

问题是，一旦你拒绝认真对待爱因斯坦的言论，你就切断了对话——一场值得继续的对话。

The problem is that once you refuse to take Einstein’s statement seriously, you cut off the conversation—a conversation that deserves to be pursued.

爱因斯坦的陈述客观上很有趣，但实际上并没有多大意义。让我们假设它是正确的。我们应该用它做什么？它能如何帮助我们？没有任何具体的细节或实用的建议，很难从中学到任何东西。

Einstein’s statement is objectively intriguing but it doesn’t really say much. Let’s assume that it is correct. What are we supposed to do with it? How can it help us? Without any concrete details or practical advice, it’s difficult to learn anything from it.

令人惊讶的是，没有人有意识地回答：“阿尔伯特，你刚才说的确实很有趣，但我们想知道更多。你能给我们解释一下吗？我们想知道秘密的细节，想知道你到底是怎么做到的。你想去喝杯咖啡吗？或者去树林里散散步？来吧，把一切都说出来；我们有很多问题要问！”

It’s kind of surprising that no one had the presence of mind to respond, “Albert, what you just said is really interesting, but we’d like to know more. Could you explain it to us? We want to know the secret details, to learn how you really do it. Do you want to go get a coffee? Or maybe go for a nice walk in the woods? Come on and tell it all; we have loads of questions!”

我本来想问的第一个问题非常愚蠢：

The first questions I would have liked to ask are pretty inane:

1. 阿尔伯特，你的好奇心从何而来？

1. Albert, where does your curiosity come from?

我不知道有多少人有足够好奇心把自己关在房间里，思考理论物理问题。但我确实认识一些人，他们都说了同一句话：如果他们把自己关在房间里研究理论物理问题，当然部分是出于科学野心，但主要是因为他们从中获得了真正的乐趣。

I don’t know many people curious enough to shut themselves in a room and meditate upon problems in theoretical physics. But I do know a few, and they all say the same thing: that if they shut themselves in a room and study problems in theoretical physics, it’s of course in part scientific ambition, but mostly because they get real pleasure from it.

所以问题就变成了：阿尔伯特，你如何从学习物理中获得乐趣？

So the question becomes: Albert, how do you derive pleasure from studying physics?

2. 你怎样做才能不灰心？

2. How do you keep from getting discouraged?

充满好奇心意味着能够对事物产生浓厚的兴趣，并保持坚定不移的决心和永不放弃的毅力。爱因斯坦显然找到了一种在别人失败时不放弃的秘诀。他的秘诀是什么？

To be passionately curious means to have the ability to be interested in things with an unwavering commitment, with an intensity and tenacity that never fails. Einstein clearly found a secret means of not giving up where others had faltered. What was his secret?

从事纯数学研究让我学到了一个必不可少的事情是这样的：当你把自己关在房间里解决一个难题时，你只有一个愿望：尽快离开那里。

Doing research in pure mathematics has taught me one essential thing: that when you shut yourself in a room with a difficult problem, you have only one wish: to get out of there as fast as you can.

达到智力的极限，徒劳地前进，奋斗数月，感到自己太愚蠢而无法理解，也不知道如何度过难关，这真是太可怕了。

It’s simply terrifying to reach the limits of your intelligence, to push on in vain, to struggle for months, to feel too stupid to understand and have no idea how to pull through.

爱因斯坦找到了一种方法来克服恐惧并抵制逃跑的冲动。他是怎么做到的？

Einstein found a way to tame his fears and resist the impulse to flee. How did he do it?

3. 当你独自一人待在房间里遇到问题时，到底发生了什么事？

3. When you’re alone in a room with a problem, what exactly is going on?

或者更明确地说：爱因斯坦是如何解决这个问题的？他是如何得到它的？他玩弄了什么？

Or, to be more explicit: what did Einstein do with the problem? How did he get his hands on it? What was he playing around with?

使用如此琐碎的语言似乎有些愚蠢，但说实话：我们真正想知道的是有趣的细节。我们想知道爱因斯坦的脑子里到底在想什么。我们想知道他实际上是怎么做到的。我们想知道爱因斯坦的技巧，他的每次都奏效的秘密魔法。

It might seem silly to use such trivial language, but let’s be honest: what we really want to know are the juicy details. We want to know what really went on inside Einstein’s head. We want to know how he actually did it. We want to know Einstein’s technique, his secret magic that worked every time.

我们知道，智力创造力不只是你付出多少努力的问题。我们知道其中一定有其他东西，一种秘密成分，一种在学校里从未提及的神秘事物。

We know that intellectual creativity isn’t just a question of how much work you do. We know there has to be something else, a secret ingredient, something mysterious that’s never even mentioned at school.

如果爱因斯坦花时间教我们他取得伟大科学发现的方法，他对人类的贡献将远远超过他在物理学方面的工作。俗话说，给人鱼，明天又饿了；教人如何钓鱼，一生都有食物。

If Einstein had taken the time to teach us his method for achieving great scientific discoveries, his contribution to humanity would have greatly surpassed his work in physics. As the saying goes, give someone a fish and they’re hungry again tomorrow; teach them how to fish and they’ll have food for a lifetime.

但这场讨论从未发生过。也永远不会发生。阿尔伯特·爱因斯坦于 1955 年 4 月 18 日在普林斯顿大学医学中心去世。进行尸检的医生自己也非常渴望发现爱因斯坦天才的秘密，以至于在不知情的情况下在家人的护送下，他取出了大脑并将其切成了千块。

But this discussion never took place. It never will take place. Albert Einstein died on April 18, 1955, at the University Medical Center of Princeton. The doctor who performed the autopsy was himself so eager to discover the secret of Einstein’s genius that, without the consent of the family, he removed the brain and sliced it into thousands of pieces.

他并没有从中学到很多东西。

He didn’t learn much from it.

方法

The Method

然而，这个问题远远超出了爱因斯坦的范畴。它已经持续了几个世纪。它关系到我们的错误信念、我们对智力和创造力的误解，以及这些错误信念对我们的限制程度。

This problem, however, goes far beyond Einstein. It’s gone on for centuries. It concerns our false beliefs, our misconceptions about intelligence and creativity, and the extent to which these false beliefs limit us.

理解爱因斯坦作品最困难的地方是数学形式主义。这也是爱因斯坦自己最大的麻烦，正如他对一位向他寻求建议的高中生承认的那样：“不要担心你在数学上的困难。我可以向你保证，我的困难更大。”

The most difficult thing about understanding Einstein’s work is mathematical formalism. It’s also what caused Einstein himself the greatest trouble, as he admitted to a high school student who asked him for advice: “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

四百年前，当时最伟大的数学家在一本后来广为人知的书中讲述了自己的一生。他的信息从一开始就非常明确。可以总结如下：“我并不比别人聪明。我只是有机会发现一种神奇的方法，让我变得比任何人都优秀。我会告诉你我是如何做到的。”

Four hundred years ago the greatest mathematician of the time talked about his life in a book that has since become famous. His message is perfectly clear from the outset. It can be summed up as follows: “I am not any more intelligent than the others. I simply had the chance to discover a magical method that allowed me to become better than anyone else. I will tell you how I did it.”

同样的下意识反应让我们难以认真对待爱因斯坦的话，也使我们无法理解这位数学家（勒内·笛卡尔）试图告诉我们什么，并且使我们无法将他的书（《方法论》）放在它应在的位置：即自我提升部分。

The same knee-jerk reaction that makes it difficult to take Einstein’s words seriously also keeps us from understanding what this mathematician (René Descartes) is trying to tell us, and keeps us from placing his book (Discourse on Method) where it belongs: in the self-improvement section.

人们一致认为，成为一名伟大的数学家并没有什么方法，就如同喝奶昔无法减肥，每周在家工作两小时也无法致富一样。

The consensus is that there isn’t any method for becoming a great mathematician, any more than there is for losing weight by drinking milkshakes or getting rich by working from home for two hours a week.

尽管笛卡尔告诉我们的恰恰相反，但这并不重要。

Little does it matter that Descartes is telling us the exact opposite.

三个错误观念

Three False Beliefs

我们将在第 14 章中更多地讨论《方法论》，但为了理解爱因斯坦和笛卡尔试图告诉我们什么，我们必须首先摆脱对数学的三个普遍看法：

We’ll talk more about Discourse on Method in chapter 14, but in order to understand what Einstein and Descartes are trying to tell us, we have to begin by ridding ourselves of three common beliefs about mathematics:

1. 为了做数学，你需要有逻辑思维。

1. In order to do mathematics, you need to think logically.

2. 我们中的一些人天生就擅长数字，还有一些人天生就具有良好的几何直觉。不幸的是，绝大多数人对数学一无所知，而且无法改变这一现状。

2. A few of us are naturally at ease with numbers and a few others naturally have a good geometric intuition. Unfortunately, the great majority of people understand nothing about math, and can’t do anything to change that.

3. 伟大的数学家天生就拥有与我们截然不同的大脑。

3. Great mathematicians are born with a brain fundamentally different from ours.

我们最好先明确第一个问题：不，数学家不会逻辑思考。事实上，逻辑思考是完全不可能的。逻辑对思考毫无帮助。我们稍后会看到它的用途。

We may as well be clear about the first one: no, mathematicians don’t think logically. It is in fact utterly impossible to think logically. Logic doesn’t help at all with thinking. We shall see later on what it is used for.

第二种谬误确实有害。它有能力让我们无可救药地压抑自己。它实际上成功地让大多数人相信数学是一个陌生而危险的领域。对于我们每个人，包括最“有天赋”的人，它都强加了一个不可逾越的限制，即每个人“天生”拥有的数学直觉。

The second fallacy is truly toxic. It has the power to make us hopelessly inhibited. It has actually succeeded at convincing most of humanity that math is a strange and dangerous territory. For each of us, including the most “gifted,” it imposes an unsurpassable limit, that of the mathematical intuition everyone is “naturally” endowed with.

第三个误解是同一主题的简单变体：要想成为爱因斯坦或笛卡尔那样的人，你必须生来如此；你无法通过努力而达到目的。当爱因斯坦或笛卡尔告诉我们不同的观点时，他们只是在嘲笑我们。

The third misconception is a simple variation on the same theme: to be like Einstein or Descartes, you have to be born that way; you can’t get there by trying. And when Einstein or Descartes tell us differently, they’re just making fun of us.

“我们无法擅长数学”的观点是错误的，但它源于一个基本事实：数学家的魔力不是逻辑而是直觉。

This vision that we’re incapable of becoming good at math is false, but it derives from an essential truth: the magic power of mathematicians isn’t logic but intuition.

官方数学与秘密数学

Official Math vs. Secret Math

爱因斯坦喜欢谈论直觉在他的发现中的重要性。“我相信直觉和灵感，”他说，而且他说这话时非常认真。至于数学家，他们很清楚存在两种不同的数学。

Einstein liked to talk about the importance of intuition in his discoveries. “I believe in intuition and inspiration,” he said, and he was being quite serious when he said it. As for mathematicians, they know quite well that there exist two different kinds of math.

官方数学可以在教科书中找到，它以一种依赖于难以理解的符号的深奥语言以一种逻辑和结构化的方式呈现。

Official math can be found in textbooks, where it is presented in a logical and structured manner, in an esoteric language that relies on indecipherable symbols.

数学家的头脑中蕴藏着秘密数学，也被称为数学直觉。它由心理表征和抽象感觉组成，通常是视觉上的，对他们来说，这些感觉非常明显，并给他们带来极大的乐趣。但当要与世界分享这些感觉时，数学家们往往不知所措。对他们来说，原本显而易见的东西突然变得不那么明显了。

Secret math, also known as mathematical intuition, can be found in the heads of mathematicians. It consists of mental representations and abstract sensations, often visual, that are for them quite obvious, and that give them a great deal of pleasure. But when it comes to sharing these sensations with the rest of the world, mathematicians are often at a loss. What had seemed so evident to them is suddenly less so.

为了记录他们的想法，数学家不得不发明那些深奥的语言和那些难以理解的符号，就像音乐家不得不发明复杂的乐谱来记录他们的作品一样。不过音乐家有一个巨大的实际优势：他们只需要播放他们的音乐，每个人都能立即明白它的意思，而不需要解读乐谱。

To transcribe their ideas, mathematicians have had to invent that esoteric language and those indecipherable symbols, just as musicians had to invent a complex musical notation in order to transcribe their compositions. Except that musicians have one enormous practical advantage: they only have to have their music played for everyone to immediately understand what it’s about, without needing to decipher the written score.

数学家没有这个选择，这对他们来说是个大问题。在他们心中，这些想法是明亮、简单和强大的。在纸面上，它们变得呆滞和可悲。数学家的诅咒是他们只能在自己的脑海里玩数学。

Mathematicians don’t have this option and it’s a huge problem for them. In their minds, the ideas are luminous, simple, and powerful. On paper, they become stunted and sad. The mathematicians’ curse is that they can only play math in their own heads.

如果在教孩子们音乐时，给他们莫扎特或迈克尔杰克逊的乐谱让他们解读，而他们却从未听过他们的演奏，那么音乐就会像数学一样被普遍讨厌。

If you taught children music by giving them the written scores for Mozart or Michael Jackson to decipher without their ever having heard it played, music would be as universally hated as math.

直觉是数学的灵魂。没有直觉，数学就毫无意义。但你不能由此得出结论说，如果你不懂任何数学，那么你就无法改变这一点。

Intuition is the soul of mathematics. Without intuition, math becomes meaningless. But you mustn’t conclude from this that if you don’t understand anything about math, then there’s nothing you can do to change that.

错误在于相信我们的数学直觉是静态的、不可逾越的限制。我们对数学对象的直觉不是天生的。它不是固定的。只要我们遵循正确的方法，我们就可以建立它，让它一天天变得更强大。

The mistake is in believing that our mathematical intuition is a static given, an insurmountable limit. The intuition that we have of mathematical objects isn’t innate. It’s not fixed. We can build it up, make it stronger day by day, as long as we follow the right method.

数学家们很清楚，官方数学并不能说明一切。他们知道真正的目标是理解书本上的内容，看到它，感受到它。数学家每天做的事情就是发展他们的直觉，使它更丰富、更清晰、更强大。与出版物和官方著作相比，数学家的直觉是他们的杰作，是他们一生的成就。

Mathematicians are well aware that official math doesn’t tell all the story. They know that the real goal is to understand what’s in the books, to see it, to feel it. What mathematicians do on a daily basis is to develop their intuition, to make it richer, clearer, more powerful. Even more so than the publications and official works, mathematicians’ intuition is their masterpiece, their lifetime accomplishment.

这种非凡的艺术能够看到看不见的东西，感受到无法感受到的东西，能够理解最深层次的东西，直到不言而喻，而 99.9999% 的人类认为这些东西极其抽象和完全无法理解——这是数学家的伟大艺术和真正的秘密。只有掌握了这门艺术的人知道它能走多远。

This extraordinary art of seeing the unseen, of feeling what can’t be felt, of understanding at the deepest level, to the point where it becomes self-evident, what 99.9999 percent of humanity deems grotesquely abstract and utterly unintelligible—this is mathematicians’ great art and their true secret. Only those who have mastered this art know how far it can lead.

但他们是如何做到的呢？这就是本书的主题。

But how do they do it? That’s the subject of this book.

数学家的三个秘密

Three Secrets of Mathematicians

1.做数学是一项体力活动。为了能够理解你还不理解的东西，你必须在头脑中执行特定的动作。这些动作是看不见的，但不可或缺。它们旨在扩展你的直觉，发展新的、更深的、更强大的心理表征。从短期来看，这种活动可能很累，但从长远来看，它会让你变得无比强大。学习如何做数学就是学习如何利用自己的身体。这就像学习如何走路、游泳、跳舞或骑自行车。这些看不见的动作不是天生的，但我们都有学习它们的能力。

1. Doing math is a physical activity. To become capable of understanding what you don’t yet understand, you have to perform specific actions in your head. These actions are invisible yet indispensable. They aim at expanding your intuition and developing new, deeper, and more powerful mental representations. In the short run this activity can be exhausting, but in the long run it makes you incredibly stronger. Learning how to do math is learning how to make use of your own body. It’s like learning how to walk, swim, dance, or ride a bike. These unseen actions aren’t innate, but we all have the ability to learn them.

2.有一种方法可以让你擅长数学。这种方法在学校里从未教授过。它与任何学术方法都不一样，而且违背了传统的教育原则。它试图让事情变得更容易，而不是更困难。你可以把它比作冥想、瑜伽、攀岩或武术。它包括克服恐惧、克服面对未知时的逃避反射以及在被反驳中找到乐趣的技巧。这种方法的确切范围实际上比数学更广泛。这是一种重新编程我们直觉的通用方法，从这个意义上说，这是一种变得更聪明的方法。

2. There’s a way to become good at math. This method is never taught in school. It doesn’t resemble any academic method and goes against the traditional tenets of education. It tries to make things easier rather than more difficult. You can compare it to meditation, yoga, rock climbing, or martial arts. It includes techniques to overcome our fears, conquer our flight reflex in the face of the unknown, and find pleasure in being contradicted. The method’s exact scope is actually broader than math. It’s a universal method for reprogramming our intuition and, in that sense, it’s a method for becoming more intelligent.

3.伟大数学家的大脑和我们的大脑一样。毫无疑问，数学天赋就像任何其他体力活动的天赋一样，在个体之间并不是平均分布的。但这些生物学差异所起的作用远比大多数人认为的要小。

3. The brains of great mathematicians work the same way as ours. There’s no doubt that natural aptitude in math, like natural aptitude in any other physical activity, isn’t equally distributed among individuals. But these biological differences play a far lesser role than most people assume.

后一点无疑是有争议的。数学的一个显著特点是经常出现一些非常有天赋的人，他们从小就表现出远远优于同龄人的能力。另一方面，许多人在高中数学甚至小学数学方面都很吃力。在没有更好的解释的情况下，将这种极端的不平等归因于“天生”的天赋是很自然的。

The latter point is undoubtedly controversial. A striking aspect of mathematics is the regular occurrence of incredibly talented individuals who appear out of nowhere and, from a young age, demonstrate abilities vastly superior to that of their peers. At the other end of the spectrum, many people struggle with high school math or even primary school math. In the absence of a better explanation, it is natural enough to attribute this extreme level of inequality to “innate” talent.

但事实上，需要解释的能力差距对于遗传学来说太过极端。人类确实表现出先天的生物学差异，但总体而言，我们是一个相当同质的物种。人们在身高、肌肉力量、心输出量和肺活量方面存在差异，而这种差异的一部分可以追溯到遗传因素。然而，这些差异从未涵盖多个数量级。

But the competence gap that requires explaining is, in fact, too extreme for genetics. Human beings do exhibit innate biological differences, but overall we are a fairly homogeneous species. People differ in height, muscular strength, cardiac output, and lung capacity, and part of this variability can be traced to genetic factors. Yet those differences never encompass multiple orders of magnitude.

用我们将在下一章中提到的比喻来说，数学是如此不平等，就好像有些人可以跑一百米一秒钟内就能冲刺，而大多数人则需要一周才能冲刺。虽然可以想象有些人可能在基因上具有更高效、更强大的神经代谢能力，使他们的数学能力提高一倍，或者提高十倍，但很难相信仅凭基因就能解释如此荒谬的不平等程度。

To use a metaphor that we’ll develop in the next chapter, math is so unequal that it’s as if some people could run the one-hundred-meter dash in under a second, while the majority wouldn’t make it in a week. While it’s conceivable that some people may genetically be endowed with a neuronal metabolism that is more efficient and powerful, making them, let’s say, twice as capable in math, or why not ten times as capable in math, it’s hard to believe that genes alone could explain such an absurd level of inequality.

这里有一个更简单、更可信的解释。养成良好的思维习惯，采取正确的心理态度，可以让你的数学水平提高一亿倍。但是，学校从来没有教过数学好的方法。你只能通过偶然的机会达到这个目标。你只能靠自己偶然发现这个方法的片段。大多数人最终什么也没发现，因为这个方法的某些要点令人惊讶且违反直觉。人们很容易忽视它们。

Here is a simpler and much more credible explanation. Developing good mental habits, adopting the right psychological attitude, can make you a billion times better at math. But the method for becoming good at math has never been taught in schools. You can reach it only by accident. You’re left to discover, by yourself and by chance, snippets of the method. Most people end up not discovering anything, because certain essential points of the method are surprising and counterintuitive. It’s very easy to overlook them.

伟大数学家的大脑与我们大脑的工作方式相同。但他们的个人经历，以及他们发展自己对周围世界经验的方式，让他们有机会从小就熟悉这种方法。他们找到了自己的路，没有遵循既定的路径，也不知道自己在做什么，只是靠运气。

The brains of great mathematicians work in the same way as ours. But their personal history, the way in which they developed their own experience of the world around them, gave them the opportunity to familiarize themselves with the method from early childhood. They found their own way, without following a set path and without knowing what they were doing, by dumb luck.

口头传统

An Oral Tradition

数学通常被定义为对数字、形状和其他类型的抽象结构的研究。或者，有些人通过其形式方面来定义它：符号和公式、公理和定理、逻辑推理的系统使用。但一些定义足够谨慎，添加了这个奇怪的警告：没有人真正知道如何定义数学。

Mathematics is often defined as the study of numbers, shapes, and other types of abstract structures. Alternatively, some define it through its formal aspects: the symbols and formulas, the axioms and theorems, the systematic use of logical deduction. But a few definitions are careful enough to add this curious caveat: no one really knows how to define mathematics.

例如，在我写这些内容时，维基百科“数学”页面上写道“数学家们对于其学科的共同定义并没有达成普遍共识”。

For example, as I’m writing these lines, the Wikipedia page under “Mathematics” states that “there is no general consensus among mathematicians about a common definition for their academic discipline.”

然而，这本书的一个关键信息是，数学家们对做数学意味着什么以及做数学的感觉有着潜在的共识。整本书可以看作是试图记录这种未表达的共识并将其“泄露”给公众。

A key message of this book, however, is that there is a latent consensus among mathematicians about what it means to do math and what it feels like. The entire book can be read as an attempt to document this unexpressed consensus and “leak it” to the general public.

如果将这一共识转化为定义，那么它就不是根据数学研究的内容来描述数学，而是将其视为一种具有特定性质的人类活动。与此同时，数学仍然是唯一一门被普遍教授的学科，没有人就数学应该是什么达成一致，这导致了一些真正奇怪的后果。

If this consensus were to be turned into a definition, it wouldn’t characterize math in terms of what it studies, but as a human activity of a particular nature. Meanwhile, mathematics remains the only academic discipline that is universally taught without anyone having agreed on what it’s supposed to be, which leads to some truly bizarre consequences.

例如，许多数学家都谈到了他们自学成才的感觉。鉴于数学在课程中占据着突出的地位，这是一个令人吃惊的悖论。当然，他们并不是真正的自学成才，因为他们在学校学到了很多东西。但他们是自学成才的，因为最重要的东西不是在学校学到的。

For example, many mathematicians have spoken about their feeling of having been self-taught. In light of the prominent role of math in the curriculum, this is a startling paradox. Of course they’re not really self-taught, since they learned a lot in school. But they are self-taught in the sense that the most important things were not taught at school.

我是这些自相矛盾的自学者之一。我在学校学习了官方数学的基础知识。与此同时，在没有人教我的情况下，我发现了秘密数学的基础知识。

I’m one of these paradoxical autodidacts. I learned the basics of official math at school. At the same time, without anyone teaching me, I discovered the rudiments of secret math.

很长一段时间里，我都没有意识到我脑子里那些看不见的动作和擅长数学之间的关系。这只是我养成的一种习惯，一种运用想象力的特殊方式。

For a long time, I wasn’t aware of the relationship between the invisible actions I was performing in my head and being good at math. It was simply a habit I’d developed, a particular way of using my imagination.

稍后我会谈到我从小就开始进行的想象力练习。起初，它只是一些单纯的游戏。例如，我闭着眼睛在房间里走来走去，同时试图记住家具的布局，这很有趣。这与我在学校学到的东西有什么关系？

I’ll talk later about the exercises in imagination that I began to do from childhood on. At first, it was nothing more than innocent games. For example, I had fun walking around the room with my eyes closed while trying to remember the layout of the furniture. What did this have to do with what I was learning at school?

我甚至不是特别擅长。我经常撞墙。我从来没有想过这个游戏，以及其他越来越难的游戏，会请允许我从与其他人相同的水平开始，发展出特别强大的几何直觉。

I wasn’t even particularly good. I often ran into the walls. I never imagined that this game, and other increasingly difficult ones, would allow me to develop, starting out at the same level as everyone else, a particularly powerful geometric intuition.

这种几何直觉是我数学生涯中的秘密武器。我开始看到别人没有看到的东西，解决别人无法解决的问题。

This geometric intuition has been the secret weapon in my mathematical career. I began to see things that no one else had seen, to solve problems that no one else had been able to solve.

直到很久以后，在与其他数学家交谈并阅读著名数学家的故事时，我才发现我的经历并不独特。

It was only much later, talking with other mathematicians and reading the stories of famous mathematicians, that I found out my experience was not at all unique.

虽然官方知识已被记录在教科书中，但数学家的秘密艺术仍然是代代相传的口头传统。它揭示了没有人敢在书中写下的东西，因为它看起来不够严肃，因为它不是科学，因为它太像自我完善了。

While the official knowledge has been transcribed in textbooks, the secret art of mathematicians has remained an oral tradition passed down from generation to generation. It reveals what no one dares write down in books because it doesn’t seem serious enough, because it’s not science, and because it resembles self-improvement too much.

这个故事值得用简单易懂的语言来讲述，因为它与我们所有人有关，无论你是数学差还是数学天才，年轻还是年老，有艺术头脑还是科学头脑。它谈论的是我们的优势而不是弱点，我们隐藏的天赋以及我们能取得的成就。

This story deserves to be told with simple and easily understandable language, because it concerns all of us, whether you’re bad at math or a math whiz, young or old, artistically or scientifically minded. It talks of our strengths rather than our weaknesses, of our hidden talents and what we can accomplish.

数学是一场内心的探险，神秘而无声。但它是一场宇宙的探险，是一场探索人类智慧、意识和语言深处的旅程。

Math is an inner adventure, secret and silent. But it’s a universal adventure, a journey into the depths of human intelligence, consciousness, and language.

在数学家之间的私人谈话中，当周围没有其他人偷听时，他们终于可以谈论他们对事物的真实看法。

In private conversations between mathematicians, when there’s no one else around to overhear them, they can finally talk about how they really see things.

是的，数学很可怕。是的，它看起来令人费解。是的，感觉你永远都无法理解它。然而，还是有办法的。

Yes, math is scary. Yes, it can seem incomprehensible. Yes, it feels like you’ll never understand it. And yet, there’s a way to get there.

2

勺子的右侧

2

The Right Side of the Spoon

我儿子阿拉姆一岁了，他正在学用勺子吃饭。说实话，这简直是一场灾难。两分钟之内，他就能把食物弄得到处都是——墙上、头发上、到处都是。

My son Aram is one year old and he’s learning how to eat with a spoon. And, truth be told, it’s a disaster. In two minutes flat he manages to get his food all over the place—on the walls, in his hair, everywhere.

我试着帮助他。我给他的勺子装了一半，然后递给他。但他抓错了一端，也就是有食物的那一端。我告诉他应该抓另一端，也就是手柄，并向他演示如何抓。但他总是坚持抓有食物的那一端。毕竟，这很有道理，因为这是他想要的食物。只是这不是这样做的。

I try to help him. I half-fill his spoon and give it to him. But he grabs the wrong end, the end with the food in it. I tell him he should grab the other end, the handle, and I show him how to do it. But he always insists on grabbing the end with the food. It makes sense, after all, because it’s the food that he wants. Except that isn’t how it’s done.

但我并不担心。他会明白的。每个人都会明白要抓勺子的哪一端。我从未听过有人说：“勺子不是我的菜。我不明白它有什么用处。勺子真的让我很烦，所以我就不用勺子了。”

Yet I’m not really worried about it. He’ll get there. Everyone ends up understanding which end of the spoon to grab. I’ve never heard anyone say: “Spoons aren’t really my thing. I’ve never seen the point. They really get on my nerves, so I just don’t use ’em.”

人类对勺子没有任何问题。没有人讨厌勺子。勺子不讨厌任何人。它们是我们生活中遇到的第一批工具之一；我们每天都会用到它们，而且会一直用下去。起初，它们是神秘而陌生的。然后它们变得熟悉起来。很快，我们就会不假思索地使用它们，就像我们自己的手一样。在某种程度上，它们与我们自己的手并没有太大的不同：我们的大脑已经将勺子、它们的用途和可能性内化了。它们已经成为我们身体的延伸。

Humans don’t have any problems with spoons. Nobody hates spoons. Spoons don’t hate anyone. They’re among the first tools we encounter in our lives; we use them every day and we use them forever. At first, they’re mysterious and strange. Then they become familiar. Pretty soon, we’re using them without thinking about it, like our own hands. And in a way, they’re not that different from our own hands: our brain has internalized spoons, their uses and possibilities. They’ve become extensions of our own bodies.

当你知道如何用勺子吃饭时，一切都变得轻而易举。你不会，这太难了。我们已经学会了用勺子，以至于我们忘记了我们必须学会它。我们忘记了，一开始，这远非易事。

When you know how to eat with a spoon, it’s easy as pie. When you don’t, it’s immensely hard. We’ve learned to use a spoon so well that we’ve forgotten that we had to learn it. We’ve forgotten that, at first, it was far from easy.

只有当你看到婴儿尝试做这个动作时，你才会意识到这个动作的复杂性。它需要出色的手眼协调能力。仅仅抓住勺子并正确地握住它就需要一点努力。更不用说握勺子的正确方法取决于你用勺子吃什么。

The complexity of this action only become obvious once you watch a baby trying to do it. It requires excellent hand-eye coordination. Simply grabbing the spoon and holding it correctly takes a bit of work. Not to mention that the right way to hold a spoon depends on what you’re eating with it.

人类登陆月球已经有五十年了，但我们才刚刚开始学习如何编写能够用勺子吃婴儿食品的机器人程序。更不用说猕猴桃了，这完全是另一回事。

It’s already been fifty years since we’ve landed on the moon, but we’re just beginning to learn how to program robots capable of eating baby food with a spoon. And let’s not even talk about kiwi, which is a whole different ball game.

严肃的事情

The Serious Stuff

勺子只是开始。然后才是严肃的事情。我们学会穿鞋和脱鞋。我们学会刷牙和剪指甲。我们学会骑自行车和溜旱冰。我们学会剥洋葱和煮咖啡。我们学会玩电子游戏和缝纽扣。我们学会开车和给咖啡机除垢。有时一开始会有点难，但很快我们就能掌握窍门了。

Spoons are only the beginning. Then the serious stuff starts. We learn how to put on our shoes and take them off. We learn how to brush our teeth and cut our fingernails. We learn how to ride a bike and roller-skate. We learn how to peel onions and make coffee. We learn how to play video games and sew on a button. We learn how to drive and decalcify the coffeemaker. Sometimes it’s a little hard at first, but pretty soon we get the hang of it.

就像勺子或自行车一样，我们的工具最终会成为我们自身的延伸。我们不用思考就使用它们。它们改变了我们。它们增强了我们。它们成​​就了我们。没有工具，我们真的一事无成。

Like the spoon or the bicycle, our tools end up becoming extensions of our selves. We use them without thinking. They transform us. They augment us. They make us what we are. Without our tools, we really don’t amount to much.

语言是所有事物中最难学的。这是一个漫长而艰辛的过程。十八个月大的孩子几乎听不懂我们说的话。但我们还是整天都在努力学习。

Language is the most difficult thing of all to learn. It’s an incredibly long, frighteningly difficult process. At eighteen months old, hardly anything we babble is intelligible. And yet we keep on trying all day long.

这足以让你失望，但我们永远不会放弃。没有人曾经说过：“语言，这真的不是我的强项。它不值得。坦白说，它很痛苦。”

It’s enough to get you down, but we never give it up. No one ever says: “Language, that’s really not my thing. It just ain’t worth it. Frankly, it’s a pain.”

没有父母会说：“简嗑着奶嘴的样子太可爱了，让她做这么难的事让我们心碎。所以我们决定不跟她说话。”

No parent ever says: “Jane’s just so cute with her pacifier that it breaks our heart to make her do anything so hard. So we’ve decided not to speak to her.”

语言不是一种选择。它不只是上流社会、富人和天才的专利。它属于每个人。

Language isn’t an option. It’s not just something for the upper class, the rich, the geniuses. It’s for everyone.

如果你真的想标记我们第一次成为人类的日期，你可以选择我们的祖先决定给每个人语言的那一天。早在有组织的宗教或成文法出现之前，我们就选择遵循这条隐含的规则：“你必须教你的孩子语言。”

If you really wanted to mark a date when we first became human, you could pick the day when our ancestors decided to give language to everyone. Well before organized religion or codified laws, we chose to follow this implicit rule: “Thou shalt teach thy children language.”

彻底的成功

A Radical Success

近两百年来，我们做出了一项新的根本性决定：教会每个人读书写字。这一决定是如此根本，以至于很难想象如果没有这一决定，我们的世界会变成什么样子，如果像以前一样，只有一小部分人能够读书写字。

More recently, in the past two hundred years or so, we made a new fundamental decision: to teach everyone how to read and write. This decision is so foundational that it’s become difficult to imagine what our world would look like without it, if only a small minority of the population, as was the case before, was able to read.

在古埃及，由于使用象形文字，书写的艺术就如同魔法。抄写员是世袭阶层，将他们的秘密一代一代地传承下去。在中世纪的欧洲，书写是一种职业。年轻人成为僧侣，与世隔绝，一生致力于抄写手稿。

In ancient Egypt, with the use of hieroglyphics, the art of writing was akin to magic. Scribes were a hereditary caste, passing down their secrets from generation to generation. In medieval Europe, writing was a vocation. Young men became monks, shut themselves off from the world, and devoted their existence to copying manuscripts.

农民们对这一切怎么看？他们是否认为阅读和写作需要一种特殊的才能，一种他们不具备的特殊智慧？他们是否觉得被排除在书面语言之外是不公平和令人沮丧的？或者他们只是告诉自己，他们没有时间、金钱或意愿，而且无论如何也没有东西可读？

What did the peasants think of all that? Did they believe that reading and writing required a special talent, a particular form of intelligence that they didn’t have? Did they find being excluded from written language unfair and frustrating? Or did they simply tell themselves they didn’t have the time, money, or desire, and that in any case there wasn’t anything for them to read?

如今，没有人认为阅读和写作需要特殊的礼物。没有人认为它毫无用处。除少数例外，所有形式的政府，无论其宗教或意识形态信仰如何，都将初等教育视为绝对优先事项。

Today, no one thinks that reading and writing require a special gift. And no one believes it’s not of any use. With rare exceptions, all forms of government, whatever their religious or ideological beliefs, make primary education an absolute priority.

全球扫盲这一激进项目取得了巨大成功。当然，文盲并没有消失，但已经变得少得多。在几代人的时间里，人类就能够完成一项历史上无与伦比的全球认知转型计划。

The radical project of global literacy has been a huge success. Illiteracy hasn’t disappeared, of course, but it’s become much more rare. In a few generations, humanity was able to accomplish a global program of cognitive transformation without equal in history.

一场彻底的灾难

A Total Disaster

在开展全球扫盲运动的同时，政府还做出了另一项重大决定：让每个人都学习数学基础知识。如今，全世界的小学和中学有十多亿儿童学习数学。

At the same time that the great campaign of global literacy was being undertaken, another radical decision was made: teach everyone the basics of math. Today, in elementary and high schools around the world, more than a billion children study math.

这真是一场彻底的灾难。

And it’s a total disaster.

数亿儿童默默忍受痛苦。他们感觉他们什么都不懂，在完全超脱（他们认为学习数学毫无用处）和因不够聪明而感到羞辱之间摇摆不定。

Hundreds of millions of children suffer in silence. They feel like they don’t understand anything, and flip-flop between utter detachment (they see absolutely nothing useful in studying math) and the humiliating sensation of simply not being smart enough.

如果你问美国青少年最难的科目是什么，数学名列榜首，占37%。到目前为止，它也是人们最讨厌的科目。但如果你问他们最喜欢的科目是什么，数学又位居第一，占23%。对一些学生来说，它是最简单的科目。

If you ask American teenagers what’s the most difficult subject, math ranks at the top of the list, with 37 percent. It’s also, by far, the most hated. But if you ask what’s their favorite subject, math is again in first place, with 23 percent. For some students, it’s the easiest subject.

我们都知道这个奇怪的现象。这是家具的一部分，我们开始把它视为正常现象。我们发现，有些人喜欢数学，觉得它很容易，而另一些人讨厌数学，觉得它难以理解，这很正常，几乎没有人介于两者之间。

We’re all aware of this strange phenomenon. It’s part of the furniture, and we’ve come to view it as normal. We find it normal that there are people who love math and find it very easy, and others who hate it and find it incomprehensible, with practically no one in between.

我们发现这种情况是如此正常，以至于对数学的态度已成为我们文化刻板印象的一部分：皮肤不好、戴着眼镜的书呆子喜欢数学，酷女孩时尚达人讨厌数学，叛逆的高中辍学生对数学毫不在意。

We find the situation so normal that attitudes toward math have become part of our cultural stereotypes: the nerd with bad skin and glasses who loves it, the cool girl fashionista who hates it, the rebellious high school dropout who couldn’t care less about it.

这些刻板印象既愚蠢又令人反感。我认识一些辍学者，他们后来成为了伟大的数学家。高中女生有权变得漂亮、受欢迎，同时仍然热爱数学。她也有权成为一名伟大的数学家。

These stereotypes are stupid and insulting. I know dropouts who have become great mathematicians. A high school girl has the right to be pretty and popular, and still love math. She also has the right to become a great mathematician.

我们已经习惯了，但这种情况一点也不正常。这确实很奇怪。不应该发生这样的事情。

We’ve grown accustomed to it, but the situation isn’t at all normal. It’s really rather strange. It shouldn’t have happened like that.

只需将数学学习与其他基础学习类型进行比较。青少年认为不懂阅读很酷，这正常吗？认为那些不用读出每个字母就能读得很好的人一定是某种怪人？

Just compare math learning with other basic types of learning. Would it be normal for teens to think it was cool not to know how to read? To figure that those who could read well, without having to sound out each letter, necessarily are some kind of weirdos?

高中毕业班里有一半的学生不知道怎么用勺子吃饭，或者不会自己系鞋带，这样可以吗？

Would it be okay for half of a graduating high school class not to know how to eat with a spoon? Or tie their own shoelaces?

解决高中数学问题应该像打结一样简单你的鞋带，如果不是这样，那么我们教数学的方式就有问题。

Solving high school math problems should be as easy as tying your shoelaces, and if that’s not the case, then there’s something wrong with the way we’re teaching math.

两个假设

Two Hypotheses

为了解释为什么有些人擅长数学而其他人却不擅长，通常会提出两种假设。

To explain why some people are good at math and others aren’t, two hypotheses are usually floated.

首先，这只是一个动机问题。人们不擅长数学是因为他们不喜欢数学，他们不喜欢数学是因为他们看不出数学在日常生活中有什么用处。但人们真的认为历史在日常生活中有用吗？这并不会使历史变得难以理解，历史课也不会让人们陷入恐慌。你从未见过学生因为不明白什么是战争或革命而哭泣。

The first is that it’s simply a question of motivation. People aren’t good at math because they don’t like it, and they don’t like it because they don’t see how it’s useful in their day-to-day lives. But do people really think that history, for example, is useful in their everyday lives? That doesn’t make it any less intelligible, and history classes don’t throw people into a state of panic. You’ve never seen students start crying because they don’t understand what a war or a revolution is.

事实上，数学不好的学生非常清楚数学是有用的，哪怕只是为了在学校取得成功并考上一所好大学。他们并不笨。他们非常清楚，数学不好意味着他们将无法从事许多职业，包括一些最负盛名和收入最高的职业。也许他们不明白为什么数学如此重要，但他们知道数学很重要。他们感到被排斥，这给了他们讨厌数学的极好理由。

In fact, students who aren’t good at math understand very well that it’s useful for something, if only to succeed at school and get into a good university. They’re not stupid. They know very well that being bad at math means they won’t be able to go into any number of professions, including some of the most prestigious and best paid. Maybe they don’t understand why math is so important, but they know that it is. They feel excluded, which gives them an excellent reason for hating it.

第二个假设很不合理。它假设有一种神秘的智力，即数学智力，在人群中分布不均。这种解释基于生物学，假设存在某种数学基因。擅长数学的人天生如此，其他人则运气不佳。

The second hypothesis is just plain mean. It supposes that there’s a mysterious type of intelligence, mathematical intelligence, that’s unequally distributed amongst the population. This explanation is based on biology, postulating that there’s some kind of math gene. Those who are good at math are simply born that way, and the others are out of luck.

这种想法如此普遍本身就有点令人惊讶。我们现在应该学会警惕这种想法。曾经有一段时间，人们认为某些种族是天生的一些人因为在田里工作而受到谴责，而另一些人则因为拥有种植园而受到谴责。最近，有人说女性无法驾驶战斗机。如今，这些想法已被推翻。

That this idea is so widespread is somewhat surprising in itself. We should have learned by now to be wary of these kinds of ideas. There was a time when people believed that certain races were naturally made for working in the fields, while others were made for owning the plantation. More recently, it was said that women were incapable of flying fighter jets. Today, these ideas have been discredited.

如果您仍然持怀疑态度，您将在下一章中看到，您拥有精通数学所需的所有智力能力。

If you’re still skeptical, you’ll see in the next chapter that you have all the intellectual abilities necessary to be good at math.

个体之间确实存在生物学上的不平等，但它与上述例子不同。它更像这样。想象一下，你让一个高中班级跑一百米。绝大多数人都能完成比赛。有些人需要十一秒，其他人需要十三秒或十八秒。也许其中有几个人需要三十秒才能跑到那么远。

Biological inequality between individuals does exist, but it’s nothing like the examples above. It’s something more like this. Imagine you had a senior high school class run a hundred meters. The vast majority would be able to complete the race. Some would need eleven seconds, others thirteen or eighteen. And maybe it would take a few of them thirty seconds to run that far.

要解释这些差距，遗传因素只是其中一个因素，此外还有动机、营养、生活方式以及跑步者的训练量。在田径比赛中，我们的基因并不完全相同。但对于百米赛跑来说，这些遗传因素通常最多只占几秒钟。

To explain these gaps, genetics is just one factor, alongside motivation, nutrition, lifestyle, and how much training the runners did. We’re not all genetically similar when it comes to running a track race. But for a hundred-meter race, these genetic factors typically account for only a couple of seconds at most.

现在想象一下，一个高年级学生参加百米赛跑，有些人在一秒内完成比赛，但一周后，超过一半的人甚至没有完成比赛。这大概就是你在高中毕业时看到的数学技能差距。

Now imagine a senior class running a hundred-meter race in which some finish in one second, but after a week more than half haven’t even made it. This is about the kind of gap that you see in math skills at the end of high school.

你去找那些没能成功的人。有些人坐在起跑线上。他们告诉你，百米赛跑是世界上最糟糕的事。他们不知道百米赛跑在日常生活中有什么用处，他们认为体育教练是个虐待狂。

You go looking for the students who haven’t made it. Some are sitting at the starting line. They tell you that the hundred-meter is the worst thing in the world. They have no idea what use it could be in their everyday lives, and they think that the gym coach is a sadist.

你会认真地得出结论说这个原因在于遗传因素吗？

Would you seriously conclude that the explanation is genetic?

我想让你们相信，唯一可能的解释是，这一切都是一个巨大的误解。人们不擅长数学，是因为没有人花时间给他们明确的指示。没有人告诉他们数学是一项体力活动。没有人告诉他们，数学不是要学的东西，而是要做的事情。

I want to convince you that the only possible explanation is that it’s all a giant misunderstanding. People aren’t good at math because no one has taken the time to give them clear instructions. No one has told them that math is a physical activity. No one has told them that, in math, there aren’t things to learn, but things to do.

他们抓错了勺子的一端，因为没人告诉他们勺子有一个正确的一端。

They’re grabbing the wrong end of the spoon because no one has ever told them that there was a right end to it.

数学老师说的话并不是我们真正需要记住的东西。它们只是我们每个人在自己的头脑中秘密进行的看不见的行动的指令和指示。

The words spoken by the math teacher aren’t the kind of things we really need to retain. They’re simply instructions and indicators for the unseen actions that each of us has to do secretly inside our own head.

以学习历史或生物学的方式学习数学是没用的。你不妨在瑜伽课上仔细记笔记，这样你就不会忘记任何事情。如果你不练习任何呼吸练习，那就一文不值。

Studying math the same way that you study history or biology is useless. You might as well take careful notes during a yoga class so that you don’t forget anything. If you don’t practice any breathing exercises, it’s worth nothing at all.

3

思想的力量

3

The Power of Thought

想象一个圆，完美无瑕，没有任何缺陷。任何旧圆。明白了吗？

Imagine a circle, perfectly round, without any defect. Any old circle. Got it?

现实生活中，完美的圆形并不存在。当你在纸上画一个圆时，总会有轻微的瑕疵。没有什么东西是完美的圆形——自行车轮胎不是，太阳不是，水面上的涟漪也不是。

In real life, perfect circles don’t exist. When you draw a circle on paper, there are always slight defects. Nothing is ever perfectly round—not bike tires, not the sun, not ripples on the water.

但这当然不会阻止你理解我在说什么，或者想象一个完美的圆圈。

But that certainly doesn’t stop you from understanding what I’m talking about, or being able to imagine a perfect circle.

你不仅可以想象它，还可以亲眼看到它。你可以在脑海中移动它。你可以把它放大或缩小。你可以对它做任何你想做的事情。

Not only can you imagine it, you can literally see it. You can move it around in your mind. You can make it larger or smaller. You can do whatever you want with it.

这种能力可以使你看到现实生活中不存在的事物，感觉到它们就在你面前，可以在你的脑海中轻松地移动它们，就像你可以触摸到它们一样 —— 这是你的魔力之一。

This ability to see things that don’t exist in real life, to feel that they’re there, right in front of you, to move them around in your head as easily as if you could touch them—this is one of your magic powers.

它是引导你真正理解数学的起点。

It’s the starting point on the road that will lead you to really understanding mathematics.

我们令人难以置信的抽象能力

Our Incredible Capacity for Abstraction

完美的圆是一种数学抽象。如果圆看起来像是熟悉的物体，那是因为你和所有其他人一样，具有数学抽象的自然能力。

A perfect circle is a mathematical abstraction. If circles seem like familiar objects, it’s because you, like all other humans, have a natural capacity for mathematical abstraction.

但是你的抽象能力并不局限于数学。

But your capacity for abstraction isn’t limited to math.

无论你是否愿意，你都会花很多时间抽象地看待这个世界。这是你身体的生理特征。你的大脑是一台机器，它从你的感官输入中创造抽象概念并在精神上操纵它们，就像你的肺是从空气中提取氧气并将其转移到血液中的机器一样。

Whether you want to or not, you spend much of your time viewing the world abstractly. It’s a physiological characteristic of your body. Your brain is a machine for creating abstractions from your sensory inputs and mentally manipulating them, just as your lungs are machines for extracting oxygen from the air and transferring it to your blood.

这怎么可能呢？这将是第 19 章的主题，我们将在其中了解我们的大脑结构如何自然地让我们创造和操纵抽象概念。

How is it possible? That will be the subject of chapter 19, where we’ll see how the structure of our brain naturally allows us to create and manipulate abstractions.

到那时为止，即使您不完全理解这样的奇迹是如何发生的，您也必须承认您能够想象一个圆圈。

Until then, and even if you don’t fully understand how such a miracle is possible, you have to admit that you’re able to visualize a circle.

我们令人难以置信的推理能力

Our Incredible Capacity for Reason

直线和圆能相交于三点吗？

Can a straight line intersect a circle at three points?

慢慢来。这不是陷阱。只需自己决定。试着想象一条直线与圆相交的所有方式，看看它是否有任何方式可以在三个点相交。

Take your time. It’s not a trap. Just try to decide for yourself. Try to imagine all the ways a straight line can intersect a circle and see if there’s any way it can do so at three points.

不，直线不能与圆在三点相交。

No, a straight line can’t intersect a circle at three points.

答案似乎很明显？这是因为，像所有人类一样，你拥有令人难以置信的推理能力。你不仅能够想象直线和圆圈等抽象物体，还能问自己关于这些物体的抽象问题，并在脑海中操纵它们，直到找到答案。

The answer seems obvious? That’s because, like all human beings, you have an incredible capacity for reason. Not only are you capable of imagining abstract objects like straight lines and circles, but you’re able to ask yourself abstract questions about these objects and manipulate them in your head until you find the answers.

答案对你来说似乎很明显，但如果有人告诉你他们不理解，你会怎么做？

The answer seems obvious to you, but what would you do if someone told you they didn’t understand?

你可能想以“你可以看到……”开头，但这样做不行。如果一个人不理解，那是因为他们无法像你一样清楚地看到圆和直线。解释数学就是让别人看到他们从未见过的东西。

You’d probably want to start by saying “You can see that . . . ,” but that won’t work. If a person doesn’t understand, it’s because they can’t see circles and straight lines as clearly as you do. Explaining math is getting others to see things they’ve never seen before.

你用来寻找答案的推理是直观和视觉化的。在你的头脑中，它就像一幅卡通画，主角是一个圆圈和一条直线。这种推理非常有效，但很难转化为文字。文字永远无法完全表达你在头脑中看到的所有微妙之处。

The reasoning you used to find the answer is intuitive and visual. In your head, it’s a kind of cartoon where the main characters are a circle and a straight line. This type of reasoning is very effective but difficult to translate into words. Words can never fully express all the subtleties you see in your head.

通过学习数学，你可以学会如何将视觉直觉转化为严谨的证明。这永远不会是一个完美的翻译。表达一个简单的直觉需要很多文字。这一切在你的脑海里看起来都很清晰。但一旦你开始写下来，它就会显得技术性和复杂。

By studying math, you can learn how to translate your visual intuition into rigorous proofs. It will never be a perfect translation. It takes a lot of words to express a simple intuition. It all seems so clear in your head. But once you start to write it down, it seems technical and complicated.

我们令人难以置信的直觉

Our Incredible Intuition

你是唯一一个能够看清自己内心想法的人。即使这很痛苦，你也只有努力将自己的想法严谨地转化成文字和符号，才能与他人分享。这也是确保你的直觉正确的唯一方法。

You’re the only person capable of seeing what’s in your head. Even if it’s painful, it’s only by making the effort of rigorously translating your vision into words and symbols that you can share it with others. And it’s also the only way to make sure that your intuition is right.

因为有时你的直觉是错误的。

Because sometimes your intuition is wrong.

你知道这是真的，即使你不喜欢被提醒。最快激怒某人的方式就是取笑他们的体格，但表明他们的直觉是错误的几乎同样有效。一般来说，这会激发两种防御机制之一：人们要么对自己说他们是失败者，产生自卑感，停止思考，要么说他们毕竟是对的，其他人都是白痴（并停止思考）。

You know it’s true even though you don’t like being reminded of it. The quickest way to get on someone’s nerves is to make fun of their physique, but showing that their intuition is wrong is almost as good. In general, it provokes one of two defense mechanisms: people either say to themselves that they’re losers, develop an inferiority complex, and stop thinking, or they say they’re right after all and all the others are idiots (and stop thinking).

然而，还有第三种方式。当有人告诉爱因斯坦或笛卡尔他们的直觉是错误的时，他们并没有生气。他们不认为他们是白痴。他们也不认为其他人是白痴。他们的反应不同。如何？这是本书中将反复出现的核心主题之一。

There is, however, a third way. When someone told Einstein or Descartes that their intuition was wrong, they didn’t get upset. They didn’t think they were idiots. They also didn’t think that the others were the idiots. They reacted differently. How? It’s one of the central themes that will reappear throughout this book.

在学校里，当他们教你要警惕你的直觉时，他们会犯两个错误——两个阻碍你智力发展的大错误。

In school, when they teach you to be wary of your intuition, they make two mistakes—two big mistakes that hold back your intellectual development.

第一种是夸大其词。它们会让你无缘无故地激动。当然，你的直觉有时会出错，但并非总是如此。通常它是正确的。你可以让它更经常地正确。你可以训练它看得更清楚、更清晰。数学家从与你相同的角度出发，构建出一种强大而值得信赖的远见直觉。他们使用简单的方法，比如本书中教授的方法，来实现这一点。

The first is to exaggerate things. They get you all worked up over nothing. Sure, your intuition is wrong every now and then, but not always. Often it’s right. And you can make it so that it’s right more often. You can train it to see more clearly and distinctly. Starting from the same point as you, mathematicians construct a visionary intuition that is powerful and trustworthy. They get there using simple methods, like those taught in this book.

学校犯的第二个错误是长篇大论地谈论直觉的局限性，却从不提醒你它的优点。你记住的信息是直觉是不完美的。这是一个重要的信息。但学校忘记传递一个更重要的信息：你的直觉是你最强大的智力资源。从某种意义上说，它是你唯一的智力资源。

The second mistake schools make is to talk at length about the limits of intuition without ever reminding you of its strengths. The message that sticks with you is that intuition is imperfect. And that’s an important message. But schools forget to pass on an even more important message: your intuition is your strongest intellectual resource. In a sense, it’s your only intellectual resource.

这些话不是空话，我并不是想奉承你，讨好你。

These aren’t just empty words. I’m not trying to flatter you to get on your good side.

这一切背后隐藏着一个深刻的生物学事实，我们稍后会详细讨论。这也是一个你之前经历过无数次的非常实际的事实。你知道死记硬背、运用现成的方法或逐行推理并不是真正的理解。这就是为什么你永远不会完全相信逻辑论证，而你对直觉理解的东西更放心。

Behind all this is hidden a profound biological truth that we’ll talk more about later. It’s also a very practical truth that you’ve experienced a million times before. You know that learning things by heart, applying ready-made methods, or following reasoning line by line isn’t really understanding. That’s why you never have complete confidence in logical arguments and you’re much more at ease with what you understand intuitively.

想象力的礼物

The Gift of Imagination

你早就知道你的直觉很强大。你不敢大声说出来，但你暗中依赖的其实是你的直觉。

You’ve known for a long time that your intuition is powerful. You wouldn’t dare say it out loud, but it’s really your intuition that you secretly rely on.

你可能不知道的是，在所有伟大的科学革命和所有最困难的数学理论背后总存在着直觉，而这些直觉总是和你自己的直觉一样简单。

What you may not know is that behind all the great scientific revolutions and all the most difficult mathematical theories there are always intuitions, and these intuitions are always as simple as your own.

爱因斯坦之所以能提出相对论，其实只是他脑海中的一个想法，并不比让你看到直线不能在三点处与圆相交的理论复杂多少。

What allowed Einstein to come up with the theory of relativity was a mental cartoon not much more complicated than the one that let you see a straight line can’t intersect a circle at three points.

当爱因斯坦说他相信直觉时，他指的并不是天赐的与我们截然不同的特殊直觉。如果他真的这么想，他就不会说“我没有特殊天赋”了。

When Einstein said he believed in intuition, he wasn’t referring to a special form of heaven-sent intuition radically different from our own. If he’d really thought that, he wouldn’t have said, “I have no special talent.”

这有点令人不安，但你必须接受事实。爱因斯坦谈论的是日常直觉，我们每个人都有的直觉，这种直觉通常被视为幼稚，学校教我们不去相信它。爱因斯坦只是在谈论我们想象事物的能力。这是我们每个人都拥有的天赋。你可能认为这没什么大不了的，但这真的很重要，没有人能比这更厉害。

It’s a bit disconcerting, but you have to accept the facts. Einstein was talking of everyday intuition, the kind that we all have, that which is often seen as childish and that school teaches us to distrust. Einstein was simply speaking of our ability to imagine things. It’s a gift that we’re all endowed with. You might think it’s no big deal, but it’s really quite something, and no one gets anything more than that.

如果您像爱因斯坦一样，学会运用简单、幼稚的想象力，成为您那个时代最伟大的物理学家，那么您就会像他一样说，伟大的科学发现仅仅只是好奇心的问题（人们不会认真对待您）。

If, like Einstein, you’d learned to use your simple and childish imagination to become the greatest physicist of your time, you would have said, as he did, that the great scientific discoveries are simply a matter of curiosity (and people wouldn’t have taken you seriously).

即使你还没有发明相对论，你也已经做出了惊人的成就。你已经能够在脑海中描绘出一个圆圈。你已经能够在脑海中移动它。你已经能够直观地向自己证明直线不能在三点与圆相交。

And even if you haven’t invented the theory of relativity, you’ve already done astounding things. You’ve been able to picture a circle in your head. You’ve been able to move it around with your mind. You’ve been able to visually prove to yourself that a straight line can’t intersect a circle at three points.

而你所做的一切都是通过闭上眼睛并保持静止来实现的。事实上，你能够做到这一切，是借助思想的力量。

And all that you’ve done by closing your eyes and staying still. You’ve been able to do it, literally, by the power of thought.

据我们所知，这种生物能力似乎只存在于人类。如果河马也知道如何做到这一点，那么它们隐藏得很好。

To the extent of our knowledge, this biological prowess seems to be limited to humans. If hippos also know how to do it, they’re hiding it well.

如果你能做到这些，请放心：你有成为数学高手的遗传潜能和智力。从生物学角度来看，这就是所需的一切。其他因素不是遗传的，它们也由你决定。这只是一个真诚、耐心、渴望和勇气的问题。

If you’ve been able to do these things, rest assured: you have the genetic potential and the intellectual faculties to become very good at math. From the biological perspective, that’s all that’s needed. The other ingredients aren’t genetic, and they’re also at your disposal. It’s simply a matter of sincerity, patience, desire, and courage.

创造清晰有力的图像

Creating Clear and Strong Images

伟大的想法总是直观的，总是简单的。它们甚至简单得可笑。我们只真正理解那些显而易见的事情。当它不明显时，那是因为我们还没有真正理解。

The big ideas are always intuitive and always simple. They’re even ridiculously simple. We only ever really understand things that are obvious. When it’s not obvious, it’s because we haven’t really understood.

这是人类认知的普遍规律。它表明我们的科学是人类发明的，而人类在最深层次上都是由相同的物质构成的。

This is a universal law of human cognition. It states that our science was invented by humans and that humans are, at the deepest level, all made of the same stuff.

伟大的发现是由那些只想理解的人做出的。他们只是想自己弄清楚事情。当他们不明白的时候，他们不会假装明白。他们会继续寻找正确的道路、正确的心理意象、正确的观察方式，直到他们明白为止。

The great discoveries are made by people who are simply trying to understand. They just want to make things clear for themselves. When they don’t understand, they don’t pretend they do. They continue to search for the right path, the right mental images, the right way of seeing, until it becomes obvious to them.

好消息是，通过这种方法，他们只能发现显而易见的事情。而对他们来说显而易见的事情，有一天对你来说也会变得显而易见。这是一个尽量不要让自己被吓倒的极好理由。

The good news is that with this method they can only discover things that are obvious. And what was obvious to them could someday become obvious to you as well. This is an excellent reason to try not to let yourself be intimidated.

这适用于所有智力问题，但对于数学更是如此。数学知识不是基于实验数据。它不需要积累百科全书知识。事实上，它完全基于明确的证明，这意味着每个结果都可以分解为一系列明显的推论。

That goes for all intellectual matters, but even more so for math. Mathematical knowledge isn’t based on experimental data. It doesn’t require amassing encyclopedic knowledge. In fact, it is entirely based on explicit proofs, which means that every result can be broken down into a succession of obvious deductions.

矛盾的是，为了使某件事变得对你来说显而易见，你首先必须构建心理表征让这一切发生的条件。一旦构建，这些心理图像可以让你立即看到它，而无需任何努力。但构建它们需要大量的时间和精力。

The paradox is that in order to get to the point where something becomes obvious to you, you first have to construct mental representations that allow that to happen. Once constructed, these mental images allow you to see it immediately and without any effort. But it takes a lot of time and effort to construct them.

不知不觉中，你已经在脑海中构建了足够好的圆形图像。要理解数学，你只需用其他物体重现你对圆形所做的操作，构建其他图像，然后结合这些图像来创建其他图像。

Without realizing it, you’ve already constructed a good enough mental image of what a circle is. To understand math, you simply have to reproduce what you managed to do with circles with other objects, construct other mental images, and then combine these mental images to create yet others.

没有人天生就拥有这些形象。没有人能够立即构建它们。构建它们的过程比你想象的要花费更多时间。对于每个人来说，这都是一个充满不确定性、反复试验、错误线索和重新开始的过程。而且它会持续你的一生。

No one is born with these images ready-made. No one is able to construct them instantly. The process of constructing them takes more time than you’d think. For everyone, it’s a matter of uncertainty, trial and error, false leads, and starting over again. And it goes on for your entire life.

无论你是否从事数学，你对世界的看法和你的心理意象都在不断发展。

Whether you do math or not, your vision of the world and your mental images are constantly evolving.

数学家的口口相传从这里开始。这不是成为超人的神奇秘诀，而是促进更好心理形象构建的简单原则。

The oral tradition of mathematicians begins here. It’s not a question of miraculous recipes for becoming superhuman, but of simple principles that foster the construction of better mental images.

这里的关键在于你是否能够重新掌控自己构建世界观的方式。

What’s at stake here is your ability to reclaim control of the way you construct your own vision of the world.

你知道，要保持健康，你需要锻炼身体、多吃水果和蔬菜、远离毒品、睡个好觉。但你能说出几条能帮助你构建强大而清晰的心理形象的基本原则吗？

You know that to stay healthy you need to exercise, eat a lot of fruits and vegetables, stay away from drugs, and get a good night’s sleep. But can you name the few basic principles that will help you construct strong and clear mental images?

这个问题从未被认真解决过。当每个人都试图让你相信你必须逻辑思考时，没有人帮助你发展你的直觉。

This subject has never seriously been tackled. When everyone was trying to make you believe that you have to think logically, no one helped you develop your intuition.

您没有采用任何方法，而是错误地认为您的直觉有时正确，有时错误，但最终您无能为力使其变得更好。

You’ve made do without any method and under the false belief that your intuition is sometimes right and sometimes wrong, but that in the end there’s nothing you can do to make it better.

在这种情况下，你居然还能学到东西，真是一个奇迹。

In this context, it’s a miracle that you’ve managed to learn anything at all.

然而，正如我们将在下一章中看到的那样，你已经做得很好了。你已经成功地培养了坚实的数学直觉。你可能认为自己数学很差，但你已经完美地吸收了数学思想，而这些思想在人类历史的 99% 时间里似乎都是天才的专利。

And yet, as we’ll see in the next chapter, you’ve done quite well. You’ve already managed to develop a solid mathematical intuition. You may think you’re terrible at math, yet you’ve perfectly assimilated mathematical ideas that, for 99 percent of human history, seemed reserved for geniuses.

4

真正的魔法

4

Real Magic

拿出十亿，然后拿走一亿，还剩下多少？

Take a billion. Then take away one. How much is left?

你真的不需要思考。你可以在脑海中看到答案：999,999,999。实际上，答案的形象化比发音更容易。

You don’t really need to think. You can see the answer in your head: 999,999,999. The answer is actually easier to picture than it is to pronounce.

这似乎很明显，但事实并非总是如此。例如，对于生活在古罗马的人来说，这根本不明显。

It seems obvious, and yet it wasn’t always like that. To someone living in ancient Rome, for example, it wouldn’t have been obvious at all.

在古典拉丁语中，十亿这个词并不存在（百万也不存在）。要传达这个想法，最简单的方法就是将其称为“一千乘以一千乘以一千”的乘积。尤利乌斯·凯撒时代的罗马人应该能够理解这一点，即使这可能会让他们有点头疼。但如果你告诉他们，你可以取这个数字，从中减去一，然后立即在脑海中想象出答案，他们就无法理解了。

In classical Latin, the word billion didn’t exist (neither did million). To communicate the idea, the easiest thing would have been to call it the product of “a thousand times a thousand times a thousand.” A Roman during the time of Julius Caesar should have been able to understand that, even if it might have given them a bit of a headache. But if you had told them that you were capable of taking this number, subtracting one from it, and picture the answer immediately in your head, they wouldn’t have been able to follow.

他们会把你当成数学天才。

They would have taken you for some kind of math whiz.

用罗马数字写出 999,999,999 会非常困难。如果罗马数字是你唯一知道的数字系统，那么 999,999,999 就不仅仅是一个你每天都不会遇到的大数字了。这是一个你甚至无法“看”到的数字。它是如此可怕，让你头晕目眩。有人可以毫不费力地立即“看”清楚它的想法是荒谬的。

You’d be hard pressed to write 999,999,999 in roman numerals. If roman numerals are the only numbering system you know, 999,999,999 is much more than a big number you don’t run into every day. It’s a number that you can’t even “look” at. It’s so terrifying that it makes your head spin. The idea that someone could instantly “see” it clearly and without any effort is absurd.

但古罗马人并没有什么极端之处。他们的人们对数字的理解确实相当先进。某些澳大利亚土著人的传统计数方式是基于身体部位。你用手指从 1 数到 5，然后向上移动到手臂：6 是手腕，7 是前臂，8 是肘部，9 是二头肌。当你数到 10（肩膀）时，你继续向上移动身体——12 是耳垂。然而，如果每个数字都需要一个对应的身体部位，你如何得到十亿呢？

But there’s nothing extreme about the ancient Romans. Their understanding of numbers was really quite advanced. The traditional way of counting among certain aboriginal Australian people is based on parts of the body. You count from 1 to 5 on the fingers, then move up the arm: 6 is the wrist, 7 the forearm, 8 the elbow, 9 the biceps. When you get to 10 (the shoulder), you keep going up the body—12 is the earlobe. Yet if each number needs a corresponding body part, how do you get to a billion?

在亚马逊地区，亚诺马米语言的数字系统更加严格：有一个词表示“一”，另一个词表示“二”，但没有表示“三”的词，只有一个基本上表示“很多”的统称。

In the Amazon, Yanomami languages have an even more restricted numeral system: there’s a word for “one” and another for “two,” but there’s no word for “three,” just a catchall word that basically means “a lot.”

对于以这种方式看待世界的人来说，发现 25 和 26 之间存在可以在一瞬间察觉到的明显区别，这必定是一种启示，就如同数学学生了解到可以精确描述多种不同无穷大尺寸时所经历的那样。

For someone who sees the world in this way, discovering that there’s a clear distinction between 25 and 26 that can be perceived in a split second must come as something of a revelation, comparable to what math students experience when they learn that there are many different sizes of infinity that can be precisely described.

彻头彻尾的骗局？

A Complete Sham?

古罗马的居民能够立即理解 XXV 和 XXVI 之间的区别。但你对大数字的敏捷反应会让他们相信你是个数学天才。这个想法让你微笑，因为你肯定知道你不是数学天才。

An inhabitant of ancient Rome would be able to grasp immediately the difference between XXV and XXVI. But your agility with big numbers would lead them to believe that you’re a math whiz. That idea makes you smile, because you know for certain that you’re no math whiz.

但你确定吗？

But are you sure about that?

如果您认为数学天才是具有超自然力量的某种变种人，如果您认为他们的头脑中有某种计算机，可以让他们使用您所知道的相同方法超快速地进行计算，那么您就错了。

If you think a math whiz is some kind of mutant with supernatural powers, if you think that they have some kind of computer in their head that lets them do calculations super quickly using the same methods that you know, then you’re wrong.

归根结底，数学天才就像圣诞老人：他们并不存在。当你认为自己看到了圣诞老人时，那绝不是真正的圣诞老人，只是一个装扮成他的人。当你认为自己看到了一个魔术师，从来都不是真正的魔术师，而都是魔术师，是懂得技巧的人，可以创造出他们拥有魔力的幻觉。

In the end, math whizzes are kind of like Santa Claus: they don’t really exist. When you think you’ve seen Santa, it’s never really Santa, just someone dressed up like him. When you think you’ve seen a magician, it’s never really a magician, it’s always an illusionist, someone who knows tricks that can create the illusion that they have magical powers.

当你以为自己看到了一个数学天才时，他实际上并不是一个真正的数学天才，他只不过是一个能够以独特的方式看待数字的人，能够将那些复杂而可怕的计算变得简单甚至显而易见。

And when you think you see a math whiz, it’s never really a math whiz, it’s always just someone who has a way of seeing numbers that turns calculations that you find complex and scary into something easy and even obvious.

事实是，我们基本上都不擅长心算，除非我们有一种直观的方式，可以彻底简化计算并“看到”结果。

The truth is that we’re all basically bad at mental calculation, except when we have an intuitive way of radically simplifying the calculation and “seeing” the result.

基于印度-阿拉伯数字的十进制系统是一种“技巧”，它让我们看到某些结果显而易见。数学天才和你之间的主要区别在于，他们的技巧比你的多，而且他们更习惯于玩弄这些技巧。

The decimal system based on Hindu-Arabic numerals is a “trick” that lets us see certain results as obvious. The main difference between a math whiz and you is that their bag of tricks is bigger than yours and they’re more used to playing with them.

真正的理解

Real Understanding

十进制数字书写系统对你来说太明显了，你甚至不记得学过它。这就像使用勺子一样。你使用它时不需要真正思考，就像它是你自己身体的延伸一样。当你看到 999,999,999 时，你以为自己直接看到了这个数字，却没有意识到你是在工具的帮助下看到了它。

The decimal system of writing numbers seems so obvious to you that you can’t even remember learning it. It’s just like using a spoon. You use it without really thinking about it, like it’s an extension of your own body. When you see 999,999,999, you think you’re seeing the number directly, without realizing that you’re seeing it with the help of a tool.

十进制书写纯粹是人类的发明。它不仅仅是一种书写系统，更是通往意识状态的一扇大门，在这种意识状态中，无论整数有多大，它们都成为具体而精确的对象。同时，整数的无限性也变得司空见惯。

Decimal writing is a purely human invention. More than simply a system of writing, it’s a door into a state of consciousness where whole numbers, however big they may be, become concrete and precise objects. At the same time, the infinitude of whole numbers becomes commonplace.

以前无法想象的事情突然变得司空见惯：这正是数学在你大脑中产生的效果。这是一种奇妙的感觉，一种极大的愉悦。

Something previously unimaginable suddenly becomes commonplace: this is exactly the type of effect mathematics produces in your brain. It’s a marvelous sensation, a great delight.

当你还是个孩子的时候，你会为能够数数而感到自豪10，然后是 20，然后是 100。这让你在课间休息时有了吹牛的资本。为了再吹牛，你一定想知道最大的数字。

When you were a child, you were proud to be able to count to 10, then 20, then 100. It gave you bragging rights at recess. In order to brag some more, you would have wanted to know the biggest number.

说实话，你对数字的认识，和那些能数到2或者5，并且坚信下一个数字，也就是许多，是最大的数字的人，差别并不大。

To tell the truth, your awareness of numbers wasn’t that far off from those people who can count to 2 or 5 and are firmly convinced that the next number, the number many, is the biggest number.

有一天，你意识到没有一个数字是最大的。即使你可能以其他方式得出这个结论，十进制书写也为你提供了一条捷径。你知道每个数字后面都跟着另一个数字。你知道如何将数字的连续性视为一个转动的计数器，并且你知道这个计数器可以无限转动。没有限制，没有一个特殊的数字使计数器停止工作。

One day, you realized that no number was the biggest. Even if you might have arrived at this conclusion some other way, decimal writing gave you a shortcut. You know that every number is followed by another. You know how to see the succession of numbers like a counter that turns, and you know that this counter can turn indefinitely. There’s no limit, there’s no special number after which the counter stops working.

然而，在人类历史的 99% 时间里，没有人能够在脑海中想象数字计数器的运转。

Yet for 99 percent of human history, no one had been able to picture a number counter turning in their head.

你脑海中转动的数字计数器是伟大数学家的集体劳动成果，他们从史前到中世纪塑造了我们今天所共享的数字形象。

The number counter turning in your head is the collective work of great mathematicians who, from prehistory until the Middle Ages, fashioned the image of numbers that we share today.

这个形象不是自然的。它不是在你出生那天刻在你身上的。它在一定程度上是任意的：我们可能选择了另一种数字书写系统，你看到的数字就会有所不同。

This image isn’t natural. It wasn’t inscribed in your body the day you were born. It’s partially arbitrary: we might have chosen another system for writing numbers, and you would see them differently.

四千多年前，巴比伦人发明了六十进制系统：他们用 60 进制而不是 10 进制来书写数字。巴比伦数学家是当时最先进的。您对小时、分钟和秒的心理印象仍然深受他们对数字的看法的影响。

More than four thousand years ago the Babylonians invented a sexagesimal system: they wrote their numbers in base 60 rather than base 10. Babylonian mathematicians were the most advanced of their time. Your mental image of hours, minutes, and seconds remains profoundly influenced by their vision of numbers.

然而，自然的是你有能力吸收抽象的数学并真正理解它们，改变你的大脑，让数学真正成为你的一部分。

What is natural, however, is your capacity to assimilate abstract mathematics and to really understand them, to modify your brain so that this math really becomes part of you.

你相信你能看到 999,999,999 这个数字。你真正做的是破译一个复杂而抽象的数学数字tation。你瞬间流利地解读了它，甚至没有意识到。整数可能不是你的母语，但你已经成为双语者了。

You believe you can see the number 999,999,999. What you’re really doing is deciphering a complex and abstract mathematical notation. You decipher it instantly, fluently, without even realizing it. Whole numbers may not be your mother tongue, but you’ve become bilingual.

成功的数学变得如此直观，以至于它不再像数学。如果这个例子对你来说很愚蠢，那正是因为你对它的理解最深。

Successful math becomes so intuitive that it no longer looks like math. If the example seems stupid to you, it’s precisely because you understand it at the deepest level.

真正的魔法并不存在

Real Magic Doesn’t Exist

在职业生涯的开始阶段，年轻的数学家常常感觉自己像个冒名顶替者。

At the start of their careers, young mathematicians often feel like imposters.

我很熟悉这种感觉，对我来说，这种感觉完全合理。我的博士论文中的结果是如此明显，以至于这几乎就像一个骗局。我的定理总是很简单，它们的证明从来没有任何真正的困难。

It’s a feeling I know well, and in my case it seemed entirely justified. The results contained in my PhD thesis were so obvious that it was almost like a trick. My theorems were always simple, and their proofs never contained any real difficulties.

我周围的每个人似乎都数学更好。他们研究的东西很深奥，远远超出了我的水平。他们写的论文非常难读，证明看起来非常复杂和技术性。如果我能理解其中的一些，那只是因为它们恰好比平时更容易。

Everyone around me seemed to be better at math. They were working on profound stuff that was way out of my league. They were writing papers that were extremely difficult to read, with proofs that seemed incredibly complex and technical. If I managed to understand a few of them, it’s only because they happened to be easier than usual.

我想知道如何做真正的数学，很难的数学。但我所能学到的只是简单的数学，适合初学者的数学。

I wanted to know how to do real math, difficult math. But all that I was able to learn was the easy math, the math for dummies.

这样说似乎有点愚蠢，但我确实花了好几年才意识到这只是一种视觉错觉。地平线跟着我移动。它始终停留在我的水平面上。

It seems silly to say this, but it really took me years to realize it was only an optical illusion. The horizon was shifting with me. It was always staying at my level.

真正的魔法并不存在。当你学会一个魔术时，它就不再具有魔力了。这可能很悲哀，但你最好习惯它。

Real magic doesn’t exist. When you learn a magic trick, it ceases being magical. That may be sad, but you’d better get used to it.

如果你发现你理解的数学太简单了，这并不是因为它简单，而是因为你理解它。

If you find that the math you do understand is too easy, it’s not because it’s easy, it’s because you understand it.

5

看不见的行动

5

Unseen Actions

例如，一位伟大的数学家出生在一个人们只知道如何数到 5 的文化中，但有一天他意识到自己可以走得更远。

A great mathematician is, for example, someone born into a culture where people only know how to count to 5, and one day realizes that you can go further than that.

没有人凭空发明了无限的数字。起初，数学思想是变化的和不确定的。你觉得你可以达到 6 或 7，但你无法清楚地表达出来，因为没有“六”或“七”的词。你有可以达到甚至更高的印象，但这种印象转瞬即逝。你并不完全相信它，你告诉自己有些事情不可能是对的。

No one invented the infinitude of numbers out of the blue. At first, mathematical ideas are shifting and uncertain. You have the feeling that you might be able to go to 6 or 7, but you aren’t able to articulate it because there are no words for “six” or “seven.” You have the impression of being able to go even further, but this impression is fleeting. You don’t completely believe it, you tell yourself something can’t be right.

当你遇到语言的限制时就会发生这种情况。

This is what happens when you run up against the limits of language.

为了表达你的感受，你必须发明新词，或者为已有的词创造新用法。只有当你找到用词语表达的方法时，稍纵即逝的印象才会不再转瞬即逝。这需要时间。词语来之不易，也不会马上出现。

In order to express what you feel, you have to invent new words, or create a new usage for words that already exist. Fleeting impressions cease being fleeting only after you find a way to pin them down with words. It takes time to get there. Words don’t come easily, and they don’t come right away.

发现的初始阶段是一种精神体验。你的思维超越了语言。世界被照亮了。你顿悟了。你看到了之前隐藏的事物。这些事物如此新颖，以至于还没有名字。

The initial phase of a discovery is a spiritual experience. You think outside of language. The world is illuminated. You have epiphanies. You see things that until then were hidden. Things so new they don’t yet have a name.

你知道我在说什么。你已经经历过这种奇妙的感觉。试着回忆一下你的第一次。那是你第一次伟大的数学发现的那一天。

You know what I’m talking about. You’ve already experienced this marvelous feeling. Try to recall your first time. It was the day of your first great mathematical discovery.

当你还是婴儿时，在你学会说话之前，你可能就玩过形状分类玩具。

When you were a baby, long before you could speak, you probably played with a shape-sorting toy.

你的父母向你展示了如何做到这一点。他们拿起一块积木，把它放在一个洞里。你想模仿他们。你拿起一块积木，试图把它放进一个洞里。但它没能成功。你用尽全力推，但它纹丝不动。

Your parents showed you how to do it. They took a block and put it in a hole. You wanted to copy them. You took a block and tried to put it in a hole. But it didn’t go. You pushed with all your strength but it didn’t budge.

这让你很生气。你的父母告诉你，你不能强行把它塞进去，你必须仔细观察形状并将它们配对：圆块在圆孔中，方形块在方孔中。看，这很容易，对吧？

That got you annoyed. Your parents told you that you couldn’t force it in, that you had to look carefully at the shapes and match them: the round block in the round hole, the square block in the square hole. See, it’s easy, right?

只是你听不懂他们在说什么。你根本就听不懂。圆形和方形这两个词对你来说毫无意义。你不仅缺乏词汇量，更糟糕的是：你缺乏形状本身。你不知道如何看待它们。圆形和方形对你来说是看不见的。

Except that you didn’t understand what they were saying. You didn’t really have any chance of understanding. The words round and square meant nothing to you. It wasn’t just the vocabulary that you lacked, it was worse: you lacked the shapes themselves. You didn’t know how to see them. Circles and squares were invisible to you.

你所能看到的只是你的父母能够把他们在洞里放了方块，而你没有。但你完全按照他们的动作做。对他们来说，这些动作有效，但对你来说，却无效。

All that you could see was that your parents were able to put the blocks in the holes and you weren’t. And yet you were following their actions exactly. For them, these actions worked, for you, they didn’t.

这一幕重复了几十次。几个月过去了；这是你年轻生命中最大的挫折。你的父母是魔术师，不是你。这不公平，也很残酷。这让你很生气。

The scene was repeated dozens of times. Months went by; it was the greatest frustration of your young life. Your parents were magicians, not you. It was unfair and cruel. It made you mad.

但你没有停止尝试。你数百次地回想这个令人羞辱的谜团。忘记羞辱吧，你想要理解。你想要知道这个秘密。

But you didn’t stop trying. You went back hundreds of times to this mystery that was so humiliating. Forget about the humiliation, you wanted to understand. You wanted in on the secret.

后来有一天，你明白了。你手里拿着一块积木，发现这块积木有特别之处，其中一个洞与这块积木有相同的共同点。你必须把积木放进这个洞里。

And then one bright day, you understood. You took a block in your hand and you noticed that this block had something particular about it, and that one of the holes had the same particular thing in common with the block. And it was into this hole that you had to place the block.

意识到这一点并不需要付出任何努力。你只是在做着一些平常的事情，这些事情甚至昨天都似乎不起作用。然后突然间答案就变得显而易见了。就像你的眼睛突然睁开了一样。

This realization didn’t take any effort. You were just in the middle of going through the usual motions, the motions that even yesterday didn’t seem to work. And then all of a sudden the answer seemed obvious. It was like your eyes were suddenly opened.

正是在这段时间，你发明了形状的概念。这不仅仅与那块积木和那个洞有关。这涉及所有积木和所有洞。每个积木都有其对应的洞，它们共享这个没有名字的非物质的东西。它每次都有效。这是魔术的秘密。

This was the period of your life when you invented the idea of shapes. It wasn’t only about that block and that hole. It was about all blocks and all holes. Each block had its corresponding hole, and they shared this immaterial thing that didn’t have a name. It worked every time. It was the secret of the magic trick.

你自己发明了形状的概念。语言带给你的不是你之外的预先存在的知识。你完全靠自己学会了如何看待形状，因为在你看到它们之前，没有人能向你解释它们是什么。后来你学会了描述形状的词语，但那是在你感知到它们之后。

You invented the idea of shapes by yourself and for yourself. It wasn’t a preexisting knowledge, outside of you, that language brought to you. You learned all by yourself to see the shapes because, before you could see them, no one could explain to you what they were. You later learned the words for the shapes, but only after your perception of them.

从这时起，你就无法阻止自己去看它们了。这变得异常简单。圆形和正方形、三角形和星星、心。其实这对你来说有点太简单了。你已经无法想象没有看到它们会是什么感觉。

From this point forward, you couldn’t stop yourself from seeing them. It became ridiculously easy. Circles and squares, triangles and stars, hearts. It’s actually a bit too easy for you. You’ve become incapable of imagining what it was like not to see them.

爱情故事

A Love Story

让我们澄清事实。

Let’s set the record straight.

当你理解了如何将积木放入洞中的秘诀时，你非常高兴。你为自己感到无比自豪。你露出了灿烂的笑容。

When you’d understood the secret of fitting the blocks into the holes, you were happy. You were immensely proud of yourself. You had a big smile.

你的父母也为你感到骄傲和高兴。毕竟，他们送你这个玩具是为了让你开心。

Your parents were also proud and happy for you. They’d given you the toy, after all, to make you happy.

也许你的父母不知道，但他们确实热爱数学。他们送给你这个玩具，是想把他们对数学的热爱传递给别人。他们成功了。如果你有孩子，你也会想送给他们这个玩具。

Maybe your parents didn’t know it, but they really loved math. In giving you this toy, they wanted to pass along their love of math. And they succeeded. If you have kids, you’ll also want to give them this toy.

在学校出现并开始学习数学之前，在我们产生压抑感和害怕被评判之前，我们都曾体验过数学的无穷乐趣。人类与数学之间有着一段漫长而深刻的爱情故事。

Before school came along and got all caught up with it, before our inhibitions and our fear of being judged came along, we all have experienced great joy in math. Between humans and mathematics, it’s been a long and profound love story.

你的开端很有希望。你对形状的发现确实是一个伟大的数学发现。我是认真的。这不是比喻。

Your beginnings were promising. Your discovery of shapes was really a great mathematical discovery. I’m being serious. That’s not a metaphor.

好吧，这是一个伟大的发现，从科学角度来看毫无意义。你只是重新发现了其他人已经知道的东西。但从你个人知识的角度来看，这是了不起的。

Okay, it was a great discovery that was entirely pointless from a scientific perspective. You’d just rediscovered something everyone else was already aware of. But from the point of view of your personal knowledge, it was spectacular.

你那天的感受正是数学家在发现新事物时的感受。数学发现同样简单、深刻且显而易见。

What you felt that day was exactly what mathematicians feel when they make a discovery. A mathematical discovery is just as simple, profound, and obvious.

在笛卡尔之前，没有人知道可以用方程来描述几何图形。在他 1637 年的论文《几何学》中，在《方法论》中，他建立了代数和几何之间的桥梁，这两个数学分支之前被认为是完全独立的。这些发现是现代笛卡尔坐标概念的起源，从那以后，任何小学生都明白这一点：你可以通过提供平面上的x和y坐标来识别平面上的一个点。很难想象在笛卡尔之前没有人“见过”笛卡尔坐标。这几乎是荒谬的，就像想象人们看不到圆形和正方形一样。

Before Descartes, no one knew that you could describe geometric figures using equations. In Geometry, his 1637 treatise, an appendix to Discourse on Method, he established a bridge between algebra and geometry, two branches of mathematics that had previously been thought of as entirely separate. These discoveries were the origin of the modern idea of cartesian coordinates, something that’s since become obvious for any schoolkid: you can identify a point in a plane by providing its x and y coordinates. It’s hard to imagine that before Descartes no one had “seen” cartesian coordinates. It’s almost absurd, like imagining people couldn’t see circles and squares.

理解数学概念就是学会发现以前看不到的事物。学会发现它们显而易见。提升你的意识状态。

Understanding a mathematical notion is learning to see things that you could not see before. It’s learning to find them obvious. It’s raising your state of consciousness.

当你观察这个世界时，你不禁会辨别出形状、大小、纹理和颜色。但你还可能看到许多其他东西。还有其他结构、其他类型的形状、物体之间的其他类型的关系。即使你现在可能看不到它们，这些形状和结构最终也会变得显而易见。

When you look at the world, you can’t help but recognize shapes, size, textures, colors. But there are many other things you might see. There are other structures, other types of shapes, other types of relations between objects. Even if you might not be able to see them now, these shapes and structures could eventually become obvious to you.

他们距离并不太远。

They’re not that far away.

它们并不难被发现。

They’re not that difficult to see.

它们就在你的眼前。

They are literally right before your eyes.

比莉和她的朋友们

Billie and Her Friends

把合适的积木放进合适的洞里并不比用勺子吃饭难。但学会如何把合适的积木放进合适的洞里却比学会如何用勺子吃饭难得多。

Fitting the right blocks into the right holes isn’t any harder than eating with a spoon. But learning how to fit the right blocks into the right holes is a whole lot harder than learning how to eat with a spoon.

在玩勺子时，你可以通过模仿来学习。在玩积木和洞的游戏时，你试图通过模仿来学习，但没成功。你错过了关键的一步。识别积木的形状和找到正确的洞是你父母在脑海中进行的看不见的动作，而你无法直接模仿。

In the case of the spoon, you can learn by imitation. With the game of blocks and holes, you tried to learn by imitation, but that didn’t work. You were missing the critical step. Recognizing the shape of the block and identifying the right hole were unseen actions that your parents were mentally performing and that you weren’t in a position to directly imitate.

你必须记住，我们学习的大多数东西都是通过模仿而学到的。模仿的本能是普遍存在的。我们与所有其他哺乳动物都一样，而不仅仅是它们。

You have to keep in mind that most of the things we learn, we learn through imitation. The instinct for imitation is universal. We share it with all other mammals, and not only them.

我最喜欢的通过模仿学习的故事是比莉和她的朋友们的故事。比莉是一只雌性海豚，住在澳大利亚阿德莱德的波特河。比莉小时候迷路了。她与海豚群隔绝，被困在水闸里，精疲力竭，后来被救起并安置在海豚馆，直到她恢复健康。

My favorite story of learning through imitation is that of Billie and her friends. Billie was a female dolphin who lived in the Port River, Adelaide, Australia. When she was young, Billie got lost. Isolated from her group, trapped in a lock and worn out, she was rescued and placed in a dolphinarium until she recovered her health.

海豚馆里有被人类训练表演杂技的圈养海豚，Billie看到后，不由自主地开始模仿。

There were captive dolphins in the dolphinarium that humans had trained to perform acrobatic tricks. Billie saw them and spontaneously began to imitate them.

她最喜欢的绝技是尾行。海豚仰卧在水下快速游动，然后突然跃出水面。跳跃时，海豚看起来就像在用尾巴倒着走路——因此得名尾行。这是一个非常困难的动作，身体上非常强烈，除了取悦人类或在其他海豚面前炫耀之外没有其他功能。

Her favorite trick was the tail walk. That’s when a dolphin swims fast underwater, on its back, then suddenly leaps straight out of the water. With the leap, the dolphin looks like it’s walking backwards on its tail—thus the name tail walking. It’s a very difficult maneuver, physically intense, with no other function than pleasing humans or showing off in front of the other dolphins.

三周后，当比莉被放归其原生栖息地时，她继续尾随而行。这种技巧以前从未在野生海豚身上观察到过。但最有趣的是接下来发生的事情：比莉群体中的其他雌性海豚也开始做同样的事情。尾随而行在阿德莱德的海豚中风靡一时。

Three weeks later, when Billie was released back into her native habitat, she continued to tail walk. This trick had never been observed before in a wild dolphin. But the most interesting thing was what happened next: the other females in Billie’s group started to do the same thing. Tail walking became all the rage among the dolphins of Adelaide.

从这个意义上讲，我们和海豚一模一样：我们不仅能够通过观察其他海豚来学习，而且我们还有模仿它们的冲动。我们的本能促使我们相互模仿。

In this way, we’re exactly like the dolphins: not only are we able to learn by watching others, but what is more, we have an urge to imitate them. Our instinct pushes us to copy one another.

我们通过模仿学会了如何系鞋带、使用烤面包机、骑自行车。我们第一次尝试时可能做不到，但观察别人的做法可以让我们有所了解。我们或多或少知道鞋带、烤面包机或自行车的用途，也或多或少知道如何使用它们。

It’s through imitation that we learn how to tie our shoes, use a toaster, ride a bike. We may not get it right on the first try, but watching others do it gives us an idea how. We know more or less what a shoelace or toaster or bike is for, and we know more or less how to use them.

但是，由于数学依赖于看不见的动作，所以无法通过模仿来学习。

But math, because it relies on unseen actions, can’t be learned through imitation.

背越式跳高

The Fosbury Flop

要想取得数学上的发现，你必须首先为自己发明新的心理活动，在头脑中创建新的图像，而不需要事先知道如何去做或者它是否有效。

To make a mathematical discovery, you have to start by inventing for yourself new mental actions, creating new images in your head, without knowing in advance how to do it or whether it will work.

发明一种真正新颖的动作在生活中非常罕见，很难找到有据可查的历史例子。即使是迈克尔·杰克逊也不是太空步的发明者。他是通过模仿学会的。这种舞蹈动作的起源至少可以追溯到 20 世纪 30 年代，至今它的发明者仍是一位无名的天才。

Inventing a truly new action is so rare in life that it’s difficult to find well-documented historical examples. Even Michael Jackson didn’t invent the moonwalk. He learned it by imitation. The origin of the dance moves goes back at least to the 1930s, and to this day its inventor remains an anonymous genius.

迪克·福斯贝里 (Dick Fosbury) 发明了一种新动作：以他的名字命名的跳高技术。

Dick Fosbury was someone who invented a new action: the high-jump technique that bears his name.

在背越式跳高之前，主要的两种技术是交叉跳（仰卧，双腿先着地）和跨跳（俯卧，双肩先着地）。

Before Fosbury, the two main techniques were the scissor jump (on the back, legs first) and the straddle jump (on the stomach, shoulders first).

背越式跳高，背部和肩部先着地，这似乎违反直觉。脱离背景来看，没有缓冲，这似乎是自杀行为。我们的身体不想这样做。头朝下向后跳需要克服我们的本能，本能告诉我们，显然，这个动作太危险了，不能尝试。

The Fosbury flop, on the back and shoulders first, might seem counterintuitive. Taken out of context, with no cushion, it seems suicidal. Our body doesn’t want to do that. Jumping backwards headfirst requires getting over our instinct that tells us that, clearly, this move is too dangerous to be attempted.

福斯贝里并没有抄袭别人的这个动作。1963年，16岁的福斯贝里开始构思这个动作，并花了数年时间完善它。

Fosbury didn’t copy this movement from anyone else. He began to imagine it in 1963, when he was sixteen years old, and spent years perfecting it.

福斯贝里很乐意模仿别人。他并不虚荣。他并不追求原创或创新。他知道模仿是最好的学习方式，所以自然而然地，他首先尝试像其他人一样跳高。

Fosbury would have been happy to copy someone else. He wasn’t vain. He wasn’t looking to be original or creative. He knew that imitation was the best way to learn, and naturally first tried to high jump like everyone else.

他的起点是在高中，当时他是球队中最差的成员。因为他没有成功获得官方技术之后，他开始进行实验，寻找一种更聪明、更有效的跳跃方式：“我并不是想赢，而是想不输。”

His starting point was in high school, where he was the worst member of the team. Because he wasn’t successful with the official techniques, he began experimenting, looking for a more intelligent and efficient way of jumping: “It was not that I was trying to win, but I was trying to not lose.”

他的技术优势在于，他可以通过滚动越过横杆，同时重心保持在横杆下方：身体的每个部分依次越过横杆，但平均而言，身体停留在横杆下方。用同样的推力，你可以跳过更高的横杆。

The strength of his technique was that it allowed him to cross over the bar by rolling around it while his center of gravity remained under it: each part of the body successively passed over the bar but on average the body stayed below it. With the same thrust you can jump over a much higher bar.

福斯贝里深谙这些科学原理。他在大学主修土木工程。但他通过内省而非计算找到了自己的方法。他密切关注身体发出的信号，专注于那些能让他更轻松地越过横杆的动作。福斯贝里的方法既深思熟虑又深思熟虑。

Fosbury understood these scientific aspects. At university, he majored in civil engineering. But he discovered his method through introspection rather than calculation. He paid close attention to what his body was telling him, concentrating on the movements that would allow him to more easily cross the bar. Fosbury’s approach was both deliberate and meditative.

有一天，他通过调整自己的跑姿和身体姿势，将个人最好成绩提高了六英寸。这是他在学校比赛中第一次真正取得的成功。就在那时，他知道自己取得了一些成就。但他的教练们不相信他。多年来，他们一直试图说服他以“正确”的方式跳跃。福斯贝里本人并没有反驳他们。他只是说，也许他的技术不对，但对他来说是正确的。

One day, by modifying his run and the position of his body, he broke his personal best by six inches. It was his first real success in a school competition. It was then that he knew he was on to something. But his coaches didn’t believe him. For years, they continued to try to convince him to jump the “right” way. Fosbury himself didn’t really have any counterarguments for them. He just said that maybe his technique wasn’t right, but it was right for him.

凭借他的技术，福斯贝里在 1968 年墨西哥奥运会上夺得了金牌。当时他 21 岁。他的首次采访表明，他本人并不完全了解自己取得的成就：“我认为现在很多孩子都会开始尝试我的方法。我不保证结果，也不向任何人推荐我的风格。”

Thanks to his technique, Fosbury won the gold medal at the Mexico Olympics in 1968. He was twenty-one years old. His first interviews show that he himself didn’t entirely understand the depth of his achievement: “I think quite a few kids will begin trying it my way now. I don’t guarantee results, and I don’t recommend my style to anyone.”

但每个人都效仿他。从 1972 年的下一届奥运会开始，他的技术就成了常态。四十多年来，每次跳高比赛创下新纪录，都是因为福斯贝里的背越式跳高。

But everyone copied him. From the next Olympics, in 1972, and onward, his technique became the norm. For more than forty years, every time a new record is set in the high jump, it’s thanks to Fosbury’s flop.

盲目抄袭

Blindly Copying

当我们讲述数学家的实际工作方式时，福斯贝里方法中最引人注目的元素将会在本书中重现。

The most striking elements of Fosbury’s approach will reappear throughout this book, as we tell the story of how mathematicians actually work.

发现总是始于单纯而天真的理解欲望。你发明新动作不是因为你想做一些新颖和原创的事情，而是因为你不能用现有的技术达到你想要的效果。没有任何参考点，没有人指导你，你必须倾听你的身体在告诉你什么。你必须习惯以一种新的方式感受你的身体。找到解决方案意味着思考那些曾经不可想象的事情。这就像增强人类的认知能力。

A discovery always begins with the simple and innocent desire to understand. You invent new actions not because you want to do something new and original, but because you can’t get where you want to be with the existing techniques. Without any reference point, without someone to guide you, you have to listen to what your body is telling you. You have to get used to feeling your body in a new way. Finding the solution means thinking what had been unthinkable. It’s like augmenting the cognitive capacity of human beings.

数学的一个特殊之处是，理解一项发现几乎与发现本身一样具有挑战性。为了重现未曾见过的行为，你无法避免自省。你必须倾听自己，在自己内心和为自己重新发明行为。

A particularity about mathematics is that understanding a discovery is almost as challenging as making the discovery itself. In order to reproduce unseen actions, you can’t avoid introspection. You have to listen to yourself, and reinvent the actions within yourself and for yourself.

为了说明这一点，想象一个隐形的跳高比赛，在没有观众或摄像机的情况下，在一个空荡荡的房间里进行，比赛由电子设备评判，该设备验证是否越过横杆，但不记录跳高运动员的技术。

To illustrate this, imagine an invisible version of the high jump that’s done without an audience or cameras, in an empty room, with the competition judged by electronic equipment that verifies the bar was crossed without recording the jumper’s technique.

福斯贝里该如何讲述他的故事呢？

How would Fosbury have been able to tell his story?

每个人都会相信，他天生就比别人跳得高。如果他说：“我没有什么特殊才能。我只是充满好奇心”，或者“我并不是想赢，而是想不输”，没有人会相信他。

Everyone would have been convinced that he was genetically programmed to jump higher than anyone else. No one would have believed him if he’d said, “I have no special talent. I am only passionately curious,” or “It was not that I was trying to win, but I was trying to not lose.”

他可能已经写了一本书来描述他的技术和他对此的感受。但如何找到合适的词语呢？

He might have written a book describing his technique and how it felt to him. But how to find the right words?

福斯贝里说，当他第一次看到视频时，他自己也感到很惊讶，很难相信自己在镜头前看到的是这样的。屏幕上的画面在物理上是可行的，而且与他所做的动作非常吻合。对于从未见过背越式跳高的人来说，学习背越式跳高几乎和自己发明一样难。即使有详细的书面说明，也不容易。“仰卧跳起，头朝下。”真的吗？为什么要在开始前写这么多关于助跑轨迹和接近横杆时身体轴线扭转的说明？为什么要用这么多技术术语？真的有必要吗？

The first time one of his jumps was filmed, Fosbury said he was surprised himself, finding it hard to believe that what he saw on the screen was physically possible and really corresponded to what he’d done. For someone who’s never seen it done, learning to jump like Fosbury is almost as hard as inventing it yourself. Even with detailed written instructions, it’s not easy. “Jump up in the air on your back, headfirst.” Seriously? And why all these preliminary pages about the trajectory of the run-up and the twisting of the body axis at the approach to the bar? Why all this technical language? Is it really necessary?

要真正学会一种动作，你必须用语言去理解它。你必须用自己的身体去感受它，发现它自然而直观。

To truly learn a movement, you have to understand it beyond words. You have to feel it within your own body, and find it natural and intuitive.

看不见的行动

Unseen Actions

数学是神秘而困难的，因为你看不到别人是如何做的。你可以看到他们在黑板上或纸上写的东西，但你看不到他们之前在头脑中进行的操作，这些操作使他们能够思考和写下这些东西。

Math is mysterious and difficult because you can’t see how others are doing it. You can see what they’re writing on the blackboard or on a sheet of paper, but you can’t see the prior actions they performed in their heads that enabled them to think and write those things.

数学本身很简单，但让我们理解数学的心理活动却很微妙，违反直觉。这些活动是看不见的。我们不能简单地模仿别人的做法。我们缺乏足够的语言来解释如何做到这一点，而且无论如何，语言总是无法表达重点：我们内心真正的感受。

Math itself is simple but the mental actions that allow us to make sense of it are subtle and counterintuitive. These actions are invisible. We can’t simply imitate what others do. We lack the adequate words to explain how to do it, and in any case words will always miss the main point: what we really feel inside ourselves.

所有学生都必须盲目地自己重新创造动作。

All students must re-create the actions for themselves, blindly.

取笑数学老师太容易了，但试着设身处地为他们着想。如果一个人从未见过鞋子，而你们唯一的沟通方式就是电话，你会如何向他解释如何系鞋带？花几秒钟想象一下这个场景，你就会明白这有多难。这个想法太难了，让你头晕目眩。

It’s all too easy to make fun of math teachers, but try putting yourself in their place. How would you explain to someone how to tie their shoes if that person had never even seen shoes and your only means of communication was by phone? Take a few seconds to imagine the scene and you’ll see how hard it would be. The very idea is so difficult it makes your mind reel.

这是数学教学的实际情况，我们都处于同样的境遇。专业数学家与数学不好的人有共同点：他们都知道完全茫然无措的感受。

This is the practical reality of teaching math, and we’re all in the same boat. Professional mathematicians have this in common with people who are bad at math: they both know what it feels like to be totally at sea.

这种感觉是他们日常生活的一部分。参加研究会议的数学家们知道，事情很可能在开始的五分钟内就土崩瓦解。他们知道继续下去不会有任何好处，只会令人悲伤和羞辱，因为这些话根本没有意义。

This feeling is part of their everyday life. Mathematicians at research conferences know that things will probably fall apart in the first five minutes. They know that it won’t do any good to continue on, that it will just be sad and humiliating, because the words will simply have no meaning.

但他们知道，在理解过程中，迷失是正常的阶段。他们不会感到不安。他们不会假装理解他们无法理解的内容。他们甚至不会尝试做笔记。他们只会停止倾听。

But they know that getting lost is a normal stage in the understanding process. They won’t get upset. They won’t pretend to understand what they can’t understand. They won’t even try to take notes. They’ll simply stop listening.

如果他们真的想理解，他们会找到另一种方法。

If they really want to understand, they’ll find another way.

6

拒绝阅读

6

Refusing to Read

我不是收藏家。我不会从收集东西中获得乐趣。对于书籍也是如此，我一生中曾多次处理掉图书馆的大部分书籍，将大部分书籍赠送或出售。我只保留那些我特别喜欢的书籍。

I’m not a collector. I don’t derive pleasure from accumulating stuff. That also goes for books, and a number of times in my life I’ve gotten rid of a big part of my library, giving away or selling most of my books. I kept only those I was especially attached to.

我的数学书库相当有限：不到一百本。很少有人拥有一百本数学书，但有些数学家拥有的数学书要多得多。我在学习和职业生涯中积累了这些书。有些书是因为我认识作者，所以才送给我的。这么多年过去了，不到一百本也不算多。

My math library is rather limited: fewer than one hundred books. Not many people have a hundred math books, but some mathematicians have a lot more. I’ve accumulated these books throughout my studies and career. Some of them I was given because I knew the author. Fewer than a hundred, after all these years, isn’t all that much.

我需要的大部分书，我要么借阅，要么读电子版。我只买那些我真正喜欢、真正想拥有或我觉得特别漂亮的书。

Most of the books I’ve needed, I’ve either borrowed or read the electronic version. I’ve bought only those that I’ve really liked, that I’ve really wanted to own, or that I found especially beautiful.

我最喜欢的书之一，也是少数几本让我舍不得送人的书之一，是桑德斯·麦克莱恩 (Saunders Mac Lane) 所著的《工作数学家的范畴》。

One of my favorite books, one of the few that it would break my heart to give away, is Categories for the Working Mathematician by Saunders Mac Lane.

每次看到它，我都会暗自微笑。这本书于 1971 年首次出版，至今仍是范畴论的参考书目，范畴论是麦克·莱恩和塞缪尔·艾伦伯格在 20 世纪 40 年代发明的革命性数学结构观察和思考方式。

Every time I come across it, I smile to myself. This book, first published in 1971, remains a reference in category theory, the revolutionary way of seeing and thinking about mathematical structures that Mac Lane and Samuel Eilenberg invented in the 1940s.

二十年前，我刚完成博士学位，用在耶鲁大学担任助理教授的第一笔薪水买了它。很少有书能如此深刻地打动我。我觉得这本书精彩、明亮、鼓舞人心，而且写得非常好。

I bought it twenty years ago, just after I’d finished my PhD, with my first paycheck as an assistant professor at Yale University. There are few books that have touched me so deeply. I find it splendid, luminous, inspiring, and remarkably well written.

我从来没有读过它。

And I’ve never read it.

拉斐尔

Raphael

当我开始攻读博士学位时，尽管我已经是数学很优秀的人之一，尽管我的工作已经是创造新的数学，但我还是对现有的知识感到不知所措。

When I started working on my PhD, even though I was officially one of those who is very good at math, even though my job already was to produce new math, I felt overwhelmed by the existing knowledge.

每次打开研究文章，我都会卡在前几行。我缺少基础知识。我在第一篇文章中引用的参考文献中寻找它们，但这些参考文献并不容易阅读。所以我在参考文献的参考文献中搜索。

Every time I opened a research article, I got stuck on the first few lines. I was missing the basics. I looked for them in the references cited in the first article, but these references weren’t any easier to read. So I searched in the references to the references.

参考文献，参考文献的参考文献，参考文献的参考文献：永无止境。即使回到 20 世纪 50 年代的数学，我也发现它对我来说已经难以理解了。

The references, the references to the references, the references to the references to the references: it never ends. Even going back to the math from the 1950s, I found out that it was already incomprehensible to me.

几十年后，我埋头阅读了数千本书和数以万计的文章，却一无所知。我怎么能指望发明出原创的东西呢？

Here I was, decades later, buried under thousands of books and tens of thousands of articles, none of which I understood. How could I hope to invent something original?

有一天，我听说最近有一本书，主题对我的研究很有用，但不是核心。每个人都说这本书写得很清楚，写得很好。这让我很想读一读。

One day, I heard about a recent book on a subject that was useful—but not central—to my research. Everyone was saying that this book was very clear and well written. Which made me want to read it.

一周后，我甚至还没写到第三页。沮丧的我向朋友 Raphael Rouquier 寻求帮助，他是我和同事的一个年轻的数学天才。

After a week, I hadn’t even made it to the third page. Demoralized, I went to ask the help of my friend Raphael Rouquier, a young math prodigy I shared an office with.

他的反应深深地印在我的脑海里：“得了吧，大卫！难道没有人告诉你，你永远不应该读数学书吗？难道他们没有告诉你，这些书根本读不懂吗？”

His reaction remains engraved in my memory: “Come on, David! Didn’t anyone ever tell you that you should never read math books? Didn’t they tell you they’re impossible to read?”

敢于不读书

Daring Not to Read

不，没有人有勇气这么清楚地告诉我。

No. No one ever had the guts to tell me so clearly.

拉斐尔有些夸张：阅读数学书是可能的。但这不是自然而然的，需要付出巨大的努力，即使你数学很好。阅读一本数学书（而不仅仅是一本关于数学的书，比如你手里拿着的这本）几乎和写一本数学书一样困难。

Raphael was exaggerating: it is possible to read math books. But it’s not natural and it requires a tremendous effort, even if you’re very good at math. Reading a math book (and not just a book about math, like the one you’re holding) is almost as difficult as writing one.

这是有原因的。当你打开一本数学书时，你打开的书中最重要的单词具有你尚无法理解的含义。这种含义可能特定于这本书。要读懂它，你首先需要找到一种方法来理解这些单词。这需要你为每个单词和每组单词构建正确的心理图像，这需要付出沉重的代价。这种努力几乎与作者写这本书的努力一样激烈，它将使你对这个问题的理解几乎与作者一样好。

There’s a good reason for that. When you open a math book, you’re opening a book where the most important words have a meaning that you can’t yet understand. This meaning might be specific to this very book. To be in a position to read it, you first need to find a way to make sense of these words. This requires constructing for yourself the right mental images for each word and each group of words, which comes at a heavy price. The effort will be almost as intense as that which allowed the author to write the book, and it will lead you to understand the matter almost as well as the author does.

如果你真的想读，如果你有足够的时间，并且你选对了书，那么这一切都是值得的。准备好几个月的艰苦工作吧。这个入会仪式将改变你。在我的一生中，我只读了三四本数学书。我并不后悔花费时间和精力。它给了我意想不到的力量，就像我喝了一剂神奇的药水。这种力量至今仍伴随着我。但这种药水很难下咽。

If you really want to, if you have enough time for it and you’ve chosen the right book, it’s well worth the effort. Get ready for a few months of hard work. This initiation rite will transform you. In my lifetime I’ve really succeeded in reading only three or four math books. I don’t regret the time and effort. It gave me unexpected powers, as if I’d drunk a magic potion. This power remains with me today. But the potion was hard to swallow.

尽管拉斐尔有些夸张，但他本质上是对的。数学书不是用来读的。

Even if Raphael exaggerated a bit, he was essentially right. Math books aren’t made to be read.

拉斐尔是我遇到的第一个不惧怕数学的人。对于他来说，拿起一本 500 页的关于他一无所知的学科的书，从中间翻开完全没有问题。

Raphael was the first person I met who had never been afraid of math. It was no problem for him to take a five-hundred-page book on a subject he knew nothing about and open it right in the middle.

甚至他拿书的方式也和我不同。他没有把双手放在同一个地方。拉斐尔把书放在他的前额上他用一只手的手指抓住书的顶部，另一只手可以自由地快速翻页。他的技巧很简单。他从不从头开始，而是随心所欲地翻页。这与其说是一种阅读技巧，不如说是一种非阅读技巧。

Even his way of holding books was different from mine. He didn’t put his hands in the same place. Raphael balanced books on his forearm and held the top of the binding with the fingers of one hand, which left the other hand free to turn the pages very quickly. His technique was simple enough. He never started at the beginning, but wherever he felt like. It wasn’t a reading technique as much as a nonreading technique.

当你拿起一本数学书时，你脑子里总会有一个想法。也许你想理解你在某个地方遇到的一个想法，想知道某个陈述是否正确，或者想知道如何证明它。你真正感兴趣的可能是第 138 页的定义 7.4、第 227 页的定理 11.5，或者只是证明中的某一段落。

When you pick up a math book, you always have an idea in mind. Maybe you want to understand an idea you’ve run into somewhere, to know if a certain statement is correct or not, or to get an idea of how to prove it. What really interests you might be Definition 7.4 on page 138, Theorem 11.5 on page 227, or maybe just a particular passage in its proof.

拉斐尔教我直接翻到第 138 页或第 227 页，找出当时最让我感兴趣的四五行，而毫不犹豫地跳过这几行所依赖的大量前期材料。

What Raphael taught me to do was to go directly to page 138 or 227 and find the four or five lines that at the moment interested me the most, without having the least scruple for skipping the mountain of preliminary material these few lines supposedly depend upon.

这才是最令人困扰的。数学书应该有逻辑地组织起来，要想理解第 138 页或第 227 页，理论上你需要理解前面的内容。因此，线性阅读应该是唯一可能的阅读方式。但在实践中，这几乎是不可能的。

That’s what’s the most troubling. A math book is supposed to be organized logically, and to understand page 138 or 227, you theoretically need to have understood all that preceded it. Linear reading should therefore be the only possible way of reading. But in practice it’s next to impossible.

在你感兴趣的四五行中，可能有几个词你听不懂。如果这阻碍了你理解其余部分，你会想回过头去看看定义。没关系。或者你无论如何都会设法应付过去。那也没关系。

In the four or five lines that interest you, there might be a few words that you don’t understand. If that stops you from understanding the rest, you’ll want to go back to the definitions. That’s okay. Or you’ll manage to muddle through anyway. That’s okay too.

事实上，你应该做任何你想做的事情。你可以翻阅这本书十秒钟、一个小时或三个月——随便。基本原则是永远不要强迫自己按顺序阅读，而是按照自己的欲望和好奇心来阅读。

In fact, you should do whatever you feel like doing. You can leaf through the book for ten seconds, one hour, or three months—whatever. The underlying principle is never to force yourself to follow the pages in order, but to follow your own desire and curiosity.

书应该为我们服务，而不是反过来。如果我们试图把数学书当成“普通”书来读，如果我们让书来决定节奏，如果我们等着它来带我们走，那就永远不会成功给我们讲个故事。我们不是被动地听。我们没有耐心——坦率地说，我们只是不感兴趣。

The book should be at our service, rather than the reverse. It will never work if we try to read a math book like a “normal” book, if we let the book dictate the pace, if we wait for it to take us by the hand and tell us a story. We’re not there to listen passively. We don’t have the patience—and, frankly, we’re just not interested.

我们之所以来这里，是因为我们有具体的问题，因为有些具体的事情我们还不了解，而我们想要了解。无论如何，这本书不应该决定议程。我们才是提出问题的人。

We’re there because we have specific questions, because there are specific things we don’t yet understand and that we want to understand. At any rate the book should never dictate the agenda. We’re the ones asking the questions.

让我们面对现实吧。我们感兴趣的四五行代码很难理解，尤其是当它们位于书的中间时。我们可能需要几个小时才能理解。但数学书的每一页都同样难。所谓“简单”（但无聊）的前几页也同样难理解。

Let’s face the facts. The four or five lines that interest us will be hard to understand, especially if they’re in the middle of the book. It might take us hours. But every single page of a math book is equally hard. The so-called “easy” (but boring) preliminary pages are no less hard to understand.

最终，最让我们感兴趣的页面可能对我们来说是最不难的。首先因为我们对它感兴趣：有趣的东西要容易得多。还因为它必然与我们已经了解的东西相关——否则我们就不会对它感兴趣。

In the end, the page that interests us the most may end up being the least difficult for us. First of all because we’re interested in it: interesting things are a whole lot easier. And also because it’s necessarily tied to something we already understand—otherwise it wouldn’t interest us.

满足你的愿望是给这本书一个真正机会的唯一方法。如果你从头开始，你可能会在第二页就灰心丧气。

Following your desire is the only way of giving the book a real chance. If you start at the beginning, you run the risk of getting discouraged by page 2.

比尔·瑟斯顿

Bill Thurston

不仅仅是数学书。还有其他没人读的书。你读过烤面包机的用户手册吗？

It’s not only math books. There are other books that no one ever reads. Have you ever read the user manual for your toaster?

可能不是。打开烤面包机包装时，您可能看过一眼，但很可能从未打开过。当然，除非烤面包机出了问题，在这种情况下，您可以跳过开头，直接进入当时需要的页面。

Probably not. You’ve probably glanced at it when you unpacked your toaster, but most likely you never opened it. Except, of course, if you had a problem with the toaster, in which case you skipped the beginning and went straight to the page you needed at that precise moment.

把数学书比作烤面包机手册可能听起来像个笑话，但这确实是一个深刻的想法。这要归功于比尔·瑟斯顿。

Comparing math books to a toaster manual may seem like a joke, but it’s really a profound idea. And we owe it to Bill Thurston.

比尔·瑟斯顿出生于 1946 年，逝于 2012 年，是当代最迷人的数学家之一。他在几何学方面的工作具有非凡的深度和独创性，为格里戈里·佩雷尔曼于 2003 年证明著名的庞加莱猜想迈出了重要一步。这项工作使他于 1982 年获得菲尔兹奖。菲尔兹奖与阿贝尔奖并列为数学界最负盛名的奖项。

Bill Thurston, who was born in 1946 and died in 2012, is one of the most fascinating mathematicians of the recent era. His work in geometry, of exceptional depth and originality, constitutes a major step toward the proof of the famous Poincaré conjecture, achieved by Grigori Perelman in 2003. This work earned him the Fields Medal in 1982. Along with the Abel Prize, the Fields Medal is the most prestigious award in mathematics.

瑟斯顿是二十世纪最伟大的几何学家之一，但这还不是全部。很难用几句话概括他令人大开眼界的思想、他独特的才华和他无限的好奇心。据我所知，还没有其他顶级数学家与日本时装设计师三宅一生合作创作高级时装系列。

Thurston is among the greatest geometers of the twentieth century, but that’s not all. It’s not easy to sum up in a few words his mind-opening thought, his unique brilliance, his unlimited curiosity. To my knowledge, no other top-flight mathematician has collaborated with Japanese fashion designer Issey Miyake on the creation of an haute couture collection.

1994 年，瑟斯顿在《美国数学学会公报》上发表了一篇长达 20 页的文章，描述了他作为数学家的动机以及工作中发挥的心理过程。瑟斯顿特别提到，当他阅读自己熟悉领域的研究文章时，他并没有真正阅读它。他更喜欢专注于“字里行间的想法”。一旦他有了清晰的想法，形式主义和所有技术细节就会突然显得毫无用处和多余：“当想法清晰时，形式设置通常是不必要的和多余的——我常常觉得自己可以更容易地写出来，而不是弄清楚作者到底写了什么。”

In 1994 Thurston published a twenty-page piece in the Bulletin of the American Mathematical Society that described his motivation as a mathematician and the mental processes at play in his work. Thurston notably said how, when he reads a research article in a field he’s familiar with, he doesn’t really read it. He prefers to concentrate on “the thoughts between the lines.” Once he has a clear idea, the formalism and all the technical details suddenly seem useless and superfluous: “When the idea is clear, the formal setup is usually unnecessary and redundant—I often feel that I could write it out myself more easily than figuring out what the authors actually wrote.”

“这就像一台新烤面包机，附带一本 16 页的手册，”他继续说道。“如果你已经了解烤面包机，并且这款烤面包机看起来与你以前见过的烤面包机相似，那么你可能只需将其插入电源，看看它是否能正常工作，而不是先阅读手册中的所有细节。”

“It’s like a new toaster that comes with a 16-page manual,” he continued. “If you already understand toasters and if the toaster looks like previous toasters you’ve encountered, you might just plug it in and see if it works, rather than first reading all the details in the manual.”

这个比喻值得稍微延伸一下。瑟斯顿说，如果你已经知道什么是烤面包机，手册就毫无用处。但如果你以前从未见过烤面包机呢？手册真的有用吗？

The metaphor merits being stretched a bit. Thurston said that the manual is useless if you already know what a toaster is. But what if you’ve never seen a toaster before? Would the manual really be of any use?

您肯定会很高兴地了解到，您不应该将手指伸进烤面包机，也不应该在淋浴时使用它。但是，那些敦促您在使用前仔细阅读的刺耳警告并不能帮助您解开烤面包机的巨大谜团：它们是用来做什么的？

You’d certainly be glad to learn that you shouldn’t put your fingers inside it, and that you should never use it in the shower. But the blaring warnings that you’re urged to READ CAREFULLY BEFORE USING aren’t going to help you solve the great mystery of toasters: what are they used for?

我找不到烤面包机的说明书，但我找到了吸尘器的说明书。说明书长达 64 页，但从来没说过吸尘器的用途。换句话说，如果我们严格遵循官方文献，吸尘器就无法解释。除了少数收到“吸尘器礼物”的人（即他们无意中想出了正确的使用方法），没人知道该怎么使用它们。

I couldn’t find my toaster manual, but I did find the one for my vacuum cleaner. It’s sixty-four pages long and never says what you use a vacuum cleaner for. In other words, if we were to stick strictly to the official literature, vacuum cleaners would remain inexplicable. Nobody would know what to do with them, except the few folks who’d have received the “gift of vacuuming” (that is, they’d have accidently come up with a proper way to use them).

吸尘器的隐藏意义并不在手册中。这是我们通过口口相传而流传的秘密。

The hidden sense of vacuum cleaners isn’t found in the manual. It’s a secret we pass on by word of mouth.

适用于吸尘器的道理也适用于数学理论。但这一学习的基本法则仍被忽视，鲜为人知。

What goes for vacuum cleaners also goes for mathematical theories. But this fundamental law of learning remains neglected and little known.

非人类语言

A Language That’s Not Human

数学书不是用人类的语言写成的。这就是它们如此难读的原因。

Math books aren’t written in the language of humans. That’s what makes them so hard to read.

数学的官方语言与我们日常使用的语言不同，没有人能够完美地掌握两种语言。这种人工语言是纯粹的人类发明——无疑是我们悠久历史中最伟大的发明之一——是为了弥补我们所说的语言的弱点而发明的。

The official language of mathematics doesn’t work the same way as the language we speak every day, and no human could ever be perfectly bilingual. This artificial language is a purely human invention—undoubtedly one of the greatest inventions in our long history—conceived of to compensate for the weaknesses in the language we speak.

它的主要特点是用一种完全不同的方法取代我们通常的定义词语的方式。

Its main particularity is to replace our usual way of defining words with a radically different approach.

在日常生活中，我们从来不会正确地定义我们使用的词语。我们通过例子来学习它们。解释香蕉是什么的最好方法是展示一根香蕉。这种方法效果很好，但有时会遇到问题，其中最具体的问题就是：当一个东西只存在于你的脑海中时，你如何用手指去指代它？

In daily life we never correctly define the words we use. We learn them through examples. The best way to explain what a banana is, is to show one. This approach works well enough, but sometimes runs into issues, the most concrete of which is: when a thing exists only in your head, how do you point your finger at it?

从深层次上讲，数学是人类唯一成功的尝试，可以精确地描述我们无法用手指指向的事物。这是本书的中心主题之一，我们将多次回顾它。

At a profound level, math is the only successful attempt by humanity to speak with precision about things that we can’t point to with our fingers. This is one of the central themes of this book and we’ll come back to it a number of times.

在数学书中，最重要的段落不是定理或证明，而是定义。数学语言就像积木，单词是有定义的：也就是说，单词是由其他单词构建的，而这些单词本身之前已经定义过。当你无法用手指指出某样东西时，这是一个很好的解决方法。

In a math book, the most important passages aren’t the theorems or the proofs: they are the definitions. Mathematical language works like building blocks where words are really defined: that is to say, built from other words that have themselves been previously defined. When you can’t point your finger at something, that’s a good way of going about it.

用这种方法，词语的意义就完全被简化为定义的字面意思。词语只不过是抽象的外壳，除了定义的含义之外，没有任何意义：如果长着鼻子是大象的定义之一，那么被砍掉鼻子的大象就不再是大象了。

With this approach, the meaning of words is reduced entirely to the letter of their definition. Words are nothing but abstract shells that mean absolutely nothing outside the defined meaning: if having a trunk is part of the definition of an elephant, then an elephant whose trunk has been cut off immediately ceases to be an elephant.

这种方法被称为逻辑形式主义。它过于拘泥于细节，甚至有些荒唐。我们不想这样思考，而且我们也没有能力这样做。只有机器人和计算机才疯狂到可以这样做。

This approach is called logical formalism. It’s so anal retentive that it can be grotesque. We have no desire to think like that, and, furthermore, we’re not really capable of it. Only robots and computers are crazy enough to do that.

这是你正确地谈论看不见的东西所付出的代价。即使逻辑形式主义从根本上来说是陌生的，我们也可以学会与之互动，就像我们可以学会与机器人和电脑互动一样。它们可能会惹恼我们，我们可能会觉得它们很可笑，但我们最终会习惯它们的心理，最终，我们很高兴它们能为我们工作。

It’s the price you pay for speaking correctly about invisible things. Even if logical formalism is fundamentally foreign, we can learn to interact with it, in the same way that we can learn to interact with robots and computers. They may annoy us, we may find them ridiculous, but we end up getting used to their psychology and, at the end of the day, we’re glad to have them work for us.

学会观察

Learning to See

学习数学就是学习如何使用那些由逻辑形式主义定义的“空壳”词语，就像它们是普通词语一样。学习赋予这些词语直观而具体的含义。学习看待它们指向的物体，就好像它们就在我们眼前一样。这需要我们将在以下章节中讨论的特定技巧。

Learning math is learning how to use words that are “empty shells,” defined by logical formalism, as if they were ordinary words. It’s learning to give these words an intuitive and concrete meaning. It’s learning to see the objects they point to as if they were right there in front of our eyes. This requires particular techniques that we’ll talk about in the following chapters.

“看见”并不总是恰当的词语，因为有很多物质的东西是看不见的。糖的味道、丝绸的触感、节奏、歌曲、熟悉的气味、时间的流逝：这些都是我们不用看见就能感受到的东西。

See isn’t always the right word, since there are many material things you can’t see. The taste of sugar, the touch of silk, a rhythm, a song, a familiar scent, the passage of time: these are things we feel without seeing them.

将想象中的生理感觉与抽象概念联系起来的能力被称为联觉。有些人能看到彩色的字母。其他人则能看到一周中的各天，就好像它们位于他们周围的空间中一样。

The ability to associate imaginary physical sensations with abstract concepts is called synesthesia. Some people see letters in colors. Others see the days of the week as if they were positioned in the space around them.

人们普遍认为联觉是一种罕见现象，与某些精神疾病有关。但实际上，联觉是一种普遍现象，是人类认知的核心组成部分。这里有一个小测试，可以看看你是否具有联觉：看看巧克力这个词，你能感觉到声音、颜色、味道吗？看到“999,999,999”，你有没有感觉到它很大？

There’s a widespread belief that synesthesia is rare and associated with certain mental conditions. In reality it’s a universal phenomenon and a core building block of human cognition. Here’s a little test to see if you’re capable of synesthesia: looking at the word chocolate, are you able to sense a sound, a color, a taste? Looking at “999,999,999,” do you get the feeling of something large?

很少有人意识到自己的联觉能力，并尝试系统地发展它，而我们的文化不会强迫你这样做。秘密数学是一种心理瑜伽，其目标是重新控制我们的联觉能力。

What is rare, and what our culture doesn’t push you to do, is to be aware of your capability for synesthesia and to try to develop it systematically. Secret math is a mental yoga whose goal is to retake control over our ability for synesthesia.

我所说的一切都不应该让你感到惊讶，因为这对你来说并不是什么新鲜事。当你学会“看到”数字 999,999,999 而不是纸上的墨水时，这要归功于你对这种精神瑜伽的掌握。

Nothing I’m saying should surprise you, since nothing about it is new to you. When you learned to “see” the number 999,999,999 rather than ink on a page, it was thanks to your command of this mental yoga.

您童年时能做的事情，今天您仍然应该能够做。

What you could do in your childhood, you should still be able to do today.

比尔·瑟斯顿是这种观察艺术的大师。在第 10 章中，我们将讨论他设法看到的一些事物。这些事物如此非凡，以至于你很难相信。

Bill Thurston was a master of this art of seeing. In chapter 10, we’ll talk about some of the things that he managed to see. It’s so extraordinary that you’ll find it hard to believe.

我们有很多东西要向他学习。

We have a lot to learn from him.

源于人类，服务于人类

By Humans, for Humans

现在，他对烤面包机的评论应该开始真正有意义了。当一个人看一本数学书时，诀窍不是像机器人那样从头到尾阅读它。诀窍是抓住“字里行间的想法”——也就是说，对所使用的词语和所描述的情况给出直观的理解。

Now his comment on toasters should start to really make sense. When a human being looks at a math book, the trick isn’t to read it from beginning to end, like a robot would. The trick is to get at “the thoughts between the lines”—that is, to give an intuitive sense to the words being used and the situations being described.

数学书不是机器人写的，也不是为机器人写的。它们是由人类写的，为人类写的。如果我们没有能力赋予它们意义，没有“字里行间的思想”，就不会有数学书，就像没有音乐就不会有乐谱一样。

Math books aren’t written by robots, for robots. They’re written by humans, for humans. Without our ability to give them meaning, without “the thoughts between the lines,” there wouldn’t be any math books, exactly for the same reason that without music there wouldn’t be any musical scores.

分享这种人类理解的最好方式是人与人之间的直接交流。这种关于数学的交流对话是通过人类语言进行的。正如瑟斯顿所说，它从来没有像只有两个人在房间里时那样高效：“一对一时，人们使用的沟通渠道远远超出了形式数学语言的范围。他们使用动作，画图和图表，制作音效并使用肢体语言。”

The best way of sharing this human understanding is direct communication between humans. This communication about mathematics is done in human language. As Thurston said, it’s never as efficient as when there are only two people in the room: “One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use actions, they draw pictures and diagrams, they make sound effects and use body language.”

瑟斯顿指出，当一个重要的新定理被证明时，该定理的解决方案通常可以在该领域的两位专家之间的私人谈话中用几分钟解释清楚，而当着一群专家的面解释同样的结果则至少需要一个小时。而要以书面形式传达结果，则需要一篇十五到二十页的论文，即使是专家也可能需要几个小时甚至几天才能理解。

When an important new theorem is proven, Thurston notes that the solution can often be explained in a few minutes during a private conversation between two specialists in the subject, whereas explaining the same result in front of an audience of specialists takes at least an hour. And to communicate the result in a written form requires a fifteen- or twenty-page paper that even a specialist may need hours, perhaps days, to understand.

从几分钟到几天是一个相当大的飞跃。更糟糕的是，这令人沮丧。

Going from minutes to days is quite a leap. And what’s worse, it’s discouraging.

神奇的触摸

The Magic Touch

数学理解并非某种可以通过触摸传递的魔力，但它看起来就像是。当你想理解一个数学概念时，最快的方法是与真正理解它的人进行公开讨论。

Mathematical understanding isn’t some kind of magic power that can be passed along through touch, but it can seem like it. When you want to comprehend a mathematical concept, the quickest way is an open discussion with someone who really understands it.

专业数学家深知这一点。他们最担心的是自己对数学的理解困难。他们和大家一样有同样的问题，但他们知道解决方案。

Professional mathematicians are well aware of this. What worries them the most is their own difficulty in understanding mathematics. They have the same problem as everyone, but they know the solution.

当我还是一名博士生时，我并没有通过读书来提高我的理解力。我职业生涯中取得的成就很大程度上要归功于我与拉斐尔的对话。我非常幸运能与一位数学如此优秀、如此慷慨地花时间的人共用一间办公室。

When I was a PhD student, I didn’t get ahead in my understanding by reading books. What I’ve been able to accomplish in my career I owe in large part to my conversations with Raphael. I was incredibly lucky to share an office with someone so good at math and so generous with his time.

拉斐尔的解释从来都不严谨。有时他们完全错了。但他们总是简单而人性化。他们赋予意义。他们让我想要更多。拉斐尔解释了什么是奥勒姆真正想表达的是，他告诉我一个想法是如何被发明出来的，以及你需要如何从“道德”的角度去理解它。

Raphael’s explanations were never rigorous. Sometimes they were plainly wrong. But they were always simple and human. They gave meaning. They made me want more. Raphael explained what a theorem really meant. He told me how an idea had been invented and how you need to understand it “morally.”

“从道德上”理解某件事意味着能够直观地向自己解释它，并引证它为真的原因：这个故事的寓意。如果数学只是一个逻辑问题，那么就不会存在这样的事情。逻辑推理没有寓意可言。

“Morally” understanding something means being able to explain it to yourself intuitively and cite the reason why it’s true: the moral of the story. If mathematics were simply a question of logic, no such thing would exist. There is no moral to draw from logical reasoning.

“道德”解释挥舞着双手，必然会留下灰色地带。它们解释了烤面包机的用途以及如何将面包放入其中，但它们从未详细说明其接线图。如果这真的是你感兴趣的，请随意转到第 138 页的定义 7.4。

“Moral” explanations, waving your hands around, necessarily leave gray areas. They explain what toasters are used for and how to put bread into them, but they never detail their wiring diagrams. If that’s really what you’re interested in, feel free to go to Definition 7.4 on page 138.

数学研究界的生活方式和社会组织反映了这种直接对话的需求。天文学家有望远镜，核物理学家有粒子加速器，而数学家也有他们伟大的科学仪器：这个仪器就是旅行。

The way of life and the social organization of the math research community reflect this need for direct conversations. Where astronomers have telescopes or nuclear physicists have particle accelerators, mathematicians also have their great scientific instrument: and this instrument is travel.

数学家们的旅行使新思想得以传播，其传播效率是其他旅行所无法比拟的。我们喜欢长期逗留。我们需要时间交谈、喝咖啡、在黑板上涂鸦，第二天醒来时再继续讨论我们想到的问题。日本数学家斋藤京二想了解我一篇文章的“字里行间的想法”，所以他经常邀请我去京都。对我来说，这让我更好地理解了他文章的“字里行间的想法”。这类旅行是数学家生活的一部分。

The trips mathematicians take allow the diffusion of new ideas with an efficiency otherwise impossible. We love long-term stays. We need time to talk, take coffee breaks, scribble on the blackboard, and pick up the discussion the next day with a question that came to us when we woke up. A Japanese mathematician, Kyoji Saito, wanted to understand the “thoughts between the lines” of one of my articles, so he often invited me to Kyoto. For my part, it allowed me to better understand the “thoughts between the lines” of his articles. These types of trips are part of a mathematician’s life.

清晰而令人生畏

Clear and Intimidating

在同一门数学课上，有人默默忍受，但可能也有另一个学生可以用简单的语言解释。为什么这种对话几乎从未发生过？

In the same math class where someone suffers in silence, there’s probably another student who could explain things in simple language. Why does this conversation almost never take place?

那些“数学不好”的人确信自己天生就容易受rior 发挥了一定的作用。他们太拘谨，无法提出正确的问题，那些真正简单的问题，那些看似愚蠢但实际上至关重要的问题。

The certainty of people “bad at math” that they’re naturally inferior plays a role. They’re too inhibited to ask the right questions, the really simple questions, the ones that seem stupid but that in fact are fundamental.

教师也难辞其咎。他们有时会助长一种错觉，认为数学可以局限于形式方程。正如数学“好”和“坏”之间有着巨大的差距一样，“好”数学老师和“坏”数学老师之间也存在着同样的差距。

Teachers share in the blame. They sometimes foster the illusion that math can be limited to formal equations. Just as there’s an incredible gap between those “good” and “bad” at math, there’s an equal gap between “good” math teachers and “bad” math teachers.

让我试着解释一下。在数学的世界里，烤面包机是拆开的。我们都必须自己在脑子里把它们组装起来。“坏”老师会把组装烤面包机的 198 个步骤背诵下来，好像故事就这样结束了。“好”老师会尽最大努力解释烤面包机是什么。他们总是看着学生的眼睛，因为从他们的眼睛里，他们就能知道他们是否理解了。

Let me try to explain. In the world of mathematics, toasters arrive disassembled. We all have to put them together in our own heads. “Bad” teachers are the ones who recite the 198 steps to assemble the toaster as if that were the end of the story. “Good” teachers do their best to explain what a toaster is. They constantly look their students in the eyes, because it’s in their eyes that they will know if they’ve understood.

让一个连面包的用途都不知道的人学会组装烤面包机的 198 个步骤简直太卑鄙了。这就像养育孩子却不给他们讲故事一样。你不能用教机器人的方式来教人类。

Inflicting the 198 steps of putting together the toaster on someone who doesn’t even know what bread is for is just plain mean. It’s like raising children without telling them stories. You can’t teach humans the same way you would teach robots.

我不认为“坏”数学老师是故意虐待狂。也许他们没有把人类理解放在数学的首位，因为他们自己没有接受过正确的教育。也许他们没有在自己的脑海中想象出烤面包机的模样。或者情况恰恰相反：当你在自己的脑海中如此清楚地看到烤面包机时，有时你会忘记并不是每个人都以同样的方式看待它。

I don’t think that “bad” math teachers are deliberately sadistic. Maybe they don’t put human understanding front and center in mathematics because they themselves haven’t been taught the right way. Maybe they don’t picture the toaster in their own head. Or maybe it’s the opposite: when you see the toaster so well in your own head, sometimes you forget that not everyone else sees it the same way.

瑟斯顿写道：“一个人的清晰心理形象对于另一个人来说却是威胁。”

“One person’s clear mental image is another person’s intimidation,” Thurston wrote.

心理意象很难分享，因为它们转瞬即逝，而且非常主观。我们的通用语言无法精确地记录它们。正是因为我们的直觉如此隐秘和不稳定，逻辑形式主义才被发明出来。

It’s hard to share mental images, as they are evanescent and profoundly subjective. Our common language is incapable of transcribing them with precision. It’s precisely because our intuition is so secretive and so unstable that logical formalism was invented.

对于瑟斯顿和大多数富有创造力的数学家来说，数学是一种感性和肉欲的体验，位于语言的上游。逻辑形式主义是使这种体验成为可能的装置的核心。数学书可能难以阅读，但我们仍然需要它们。它们是我们在寻求真正的数学时所依赖的一种工具，唯一重要的数学：秘密的数学，存在于我们头脑中的数学。

For Thurston, as for most creative mathematicians, mathematics is a sensual and carnal experience that is located upstream from language. Logical formalism is at the heart of the apparatus that makes this experience possible. Math books may be unreadable, but we nevertheless need them. They’re a device that we rely on in our quest for the true math, the only one that matters: the secret math, the one that lies in our head.

这自然而然地引出了一个问题。人们如何有勇气和意愿去写出那些难以阅读、读者根本不感兴趣、像烤面包机使用手册一样枯燥的书？他们的动机是什么？什么样的心态会催生数学创造力？

Which brings us to a natural question. How do people find the courage and desire to write books that are unreadable, that readers couldn’t care less about, and that are as dry as the user manual for a toaster? What’s their motivation? What state of mind gives birth to mathematical creativity?

这是下一章的主题。

This is the subject of the next chapter.

7

儿童姿势

7

The Child’s Pose

“我亲爱的 Serre，谢谢你慷慨地寄给我的各种论文，以及你的来信。这里没有什么新东西。我已经完成了关于同调代数的荒谬文章。”

“My dear Serre, Thanks for the various papers you’ve so generously sent me, as well as for your letter. Nothing new here. I’ve finished my ridiculous piece on homological algebra.”

这是亚历山大·格罗滕迪克在 1956 年 11 月 13 日写给让-皮埃尔·塞尔的一封信的开头。这种随意的语气有点令人惊讶，特别是当你知道塞尔和格罗滕迪克是谁，以及这封信的内容是什么的时候。

This is how Alexander Grothendieck begins a letter written to Jean-Pierre Serre on November 13, 1956. The casual tone is a bit surprising, especially when you know who Serre and Grothendieck are, and what the letter is about.

让-皮埃尔·塞尔是二十世纪最伟大的数学家之一。职业生涯并不完全由你获得的奖项来衡量，但如果你赢得了所有奖项，那就说明了一些问题。塞尔在 1954 年 27 岁时获得了菲尔兹奖，至今仍是有史以来最年轻的获奖者。该奖项仅限于 40 岁以下的数学家，长期以来，数学领域没有相当于终身成就奖的东西。这就是阿贝尔奖于 2003 年设立的原因。在该奖设立的第一年，奖项委员会肩负着重大责任：他们必须在所有在世的数学家中选出谁将获得第一个奖项。他们决定把这个奖项授予塞尔。

Jean-Pierre Serre is one of the greatest mathematicians of the twentieth century. A career isn’t entirely measured by the awards you win, but when you’ve won them all, that’s saying something. Serre won the Fields Medal in 1954 at the age of twenty-seven, and is still the youngest ever to do so. This prize is restricted to mathematicians under forty, and for a long time there wasn’t anything equivalent to a lifetime achievement award in math. This is why the Abel Prize was created in 2003. The first year of the prize, the awards committee had a great responsibility: among all living mathematicians, they had to choose who was to receive the first award. They decided to give it to Serre.

至于格罗滕迪克，他不仅仅是一位伟大的数学家。早在 2014 年去世之前，他就已经成了一个传奇人物。

As for Grothendieck, he is much more than a great mathematician. Well before his death in 2014, he’d already become something of a legend.

他是历史上为数不多的数学家之一，他的贡献不仅限于取得深远的成果或壮观的理论。格罗滕迪克发明了一种研究数学的方法，它如此丰富和新颖，就好像他改变了数学的本质。

He’s one of those rare mathematicians—one of the few throughout history—whose contributions aren’t limited to profound results or spectacular theories. Grothendieck invented a way of approaching math that was so rich and new it was as if he’d changed the very nature of mathematics.

这就解释了为什么他经常被认为是二十世纪最伟大的数学家，只要这意味着什么。

This explains why he’s often considered “the” greatest mathematician of the twentieth century, insofar as that means anything.

至于那篇“关于同调代数的荒谬文章”，是1957年发表在日本学术期刊《东北数学杂志》上的文章《同调代数的一些方面》。

As for the “ridiculous piece on homological algebra,” that was the article “Some Aspects of Homological Algebra” that appeared in 1957 in a Japanese scholarly journal, the Tohoku Mathematical Journal.

这篇文章标志着格罗滕迪克进入了后来让他成名的领域。受塞尔的影响，他刚刚开始研究代数几何。这两个年轻人开始了历史上最富有成效的数学友谊之一。格罗滕迪克后来谈到他第一次接触代数几何时的印象时说，他“突然发现自己身处一片富饶的‘应许之地’。”

This article marked Grothendieck’s entry in the field that would make him famous. Influenced by Serre, he was just getting started in algebraic geometry. The two young men began one of the most productive mathematical friendships in history. Of his first encounter with algebraic geometry, Grothendieck would later say that he had the impression of “suddenly finding himself in a sort of ‘promised land’ of luxuriant richness.”

格罗滕迪克用了 15 年的时间来绘制这片“应许之地”。写作是他的方法的核心。他甚至说“做数学首先就是写作。 ”

Grothendieck would devote fifteen years of his life to mapping out this “promised land.” Writing was at the heart of his method. He would go as far as to say that “doing mathematics is above all writing.”

这种对写作的热爱使他写给塞尔的信更加神秘。这封“荒谬的文章”只是格罗滕迪克在应许之地的冒险经历的第一个故事。他觉得这篇文章太荒谬了，以至于完成花了他一年时间完成的任务毫无意义。“这里没有什么新东西”是他用来宣布他刚刚写完一篇历史性文章的确切用词。

This passion for writing makes his letter to Serre even more enigmatic. The “ridiculous piece” is quite simply the first tale of Grothendieck’s adventures in the promised land. He found the piece so ridiculous that finishing the task that had consumed him for a year was a nonevent. “Nothing new here” are the exact words he used to announce that he’d just finished writing a historic article.

他是在开玩笑吗？可能不是。在 2018 年的一次采访中，塞雷回忆说，格罗滕迪克的怪癖之一是他完全没有幽默感：“我不记得听过他笑。你永远不能跟他开玩笑，比如关于数学的。”

Was he joking? Probably not. In a 2018 interview Serre recounted that one of Grothendieck’s peculiarities was his total lack of humor: “I can’t recall having heard him laugh. You could never joke with him, for example, about mathematics.”

在这看似矛盾的背后隐藏着关于数学工作本质的深刻真理。格罗滕迪克的超然和轻率可能看起来令人难以理解，但当你更多地了解他的思想方法时，你会发现这是完全合理的。

Hidden behind this apparent paradox lies a profound truth about the nature of mathematical work. Grothendieck’s detachment and flippancy might seem incomprehensible, but when you know more about his intellectual approach, you’ll find it perfectly coherent.

低俗笑话

A Joke in Bad Taste

每个人都知道爱因斯坦是谁，但几乎没人听说过格罗滕迪克。

Everyone knows who Einstein is, but almost no one’s heard of Grothendieck.

比较两者并不荒谬。爱因斯坦彻底改变了物理学家对空间的认识。格罗滕迪克彻底改变了数学家对空间的认识。他甚至重新发明了点的概念，并从几何角度来探讨真理的概念。

Comparing the two isn’t absurd. Einstein revolutionized physicists’ ideas about space. Grothendieck revolutionized mathematicians’ ideas about space. He even went so far as to reinvent the concept of a point, and to approach the notion of truth from a geometric perspective.

有些数学家甚至认为，将格罗滕迪克与爱因斯坦进行比较对格罗滕迪克并不公平。他们觉得爱因斯坦的工作美丽、优雅、辉煌、令人钦佩。他们说这是天才的工作。至于格罗滕迪克的工作，他们觉得它非凡、令人震惊、崇高、令人恐惧。他们说这不可能是人类的工作。格罗滕迪克的想法并不总是容易理解的，但一旦你理解了一点，就会觉得难以置信竟然有人能想出这些想法。

Some mathematicians even think that the comparison with Einstein isn’t fair to Grothendieck. They find Einstein’s work beautiful, elegant, brilliant, admirable. They say that it’s the work of a genius. As for Grothendieck’s work, they find it extraordinary, staggering, sublime, terrifying. They say that it can’t be the work of a human being. Grothendieck’s ideas are not always easy to understand, but once you understand even a bit, it seems incredible that anyone could have come up with them.

让-皮埃尔·塞尔曾说，他个人无法完成格罗滕迪克的工作，因为“这需要巨大的力量”。塞尔在提到格罗滕迪克时，谈到了“大脑的力量”，并描述了一种超自然的力量：“无论是身体上还是智力上，都是一样的。这太不可思议了。我从未见过有如此强大力量的人。我见过智力超群的人，但格罗滕迪克超越了这一点。他是一头野兽。”

Jean-Pierre Serre once said about Grothendieck’s work that he personally would have been incapable of producing it, because “it demands enormous strength.” When Serre evokes Grothendieck, he talks of “the power of his brain” and describes a supernatural force: “Physically and intellectually, it was the same. It was extraordinary. I’ve never known anyone with as much strength. I’ve met people with incredible intellectual abilities, but Grothendieck was beyond that. He was a beast.”

格罗滕迪克本人却不这么认为。他并不认为自己比别人更有天赋。这并不是他非凡创造力的源泉：“这种能力绝不是什么非凡的天赋——就像一种非凡的大脑力量（可以这么说）……这样的天赋当然是珍贵的，值得那些生来就没有‘无上’天赋的人（比如我）羡慕。”

Grothendieck himself wasn’t of the same opinion. He didn’t think that he was more gifted than anyone else. That wasn’t the source of his uncommon creativity: “This power is in no way some extraordinary gift—like an uncommon cerebral strength, (shall we say). . . . Such gifts are certainly precious, worthy of the envy of people (like me) who haven’t been blessed with them at birth, ‘beyond measure.’”

格罗滕迪克有不同的解释：“研究人员的创造力和想象力的质量来自于他的注意力的质量，以及倾听事物的声音的质量。”

Grothendieck had a different explanation: “The quality of the inventiveness and the imagination of a researcher comes from the quality of his attention, listening to the voice of things.”

这听起来几乎和我们在第一章开头提到的爱因斯坦的话一样：“我没有特殊的天赋。我只是充满好奇心。”

It almost sounds like the same words from Einstein that we started with in chapter 1: “I have no special talent. I am only passionately curious.”

但格罗滕迪克走得更远。然而，他知道没有人会相信他，因为这些声明从来都不会被认真对待：“当你敢说这样的话时，你会看到每个人的脸上，从最愚笨的人到最聪明的人，他们都确信自己很聪明，而且比普通人聪明得多，脸上都挂着同样的微笑，一部分是尴尬，一部分是心知肚明，就好像有人开了个低俗的玩笑。”

But Grothendieck went even further. He knew, however, that no one would believe him, since these kinds of declarations are never taken seriously: “When you dare say such things, you see on everyone’s face, from the dullest who are sure they’re dull, to the smartest who are certain they’re smart and well above common mortals, the same smiles, part embarrassed, part knowing, as if someone had made a joke in bad taste.”

爱因斯坦以爱开玩笑而闻名。但格罗滕迪克则不必担心：他不允许开玩笑。

Einstein had the reputation of being something of a joker. With Grothendieck, no need to worry: jokes were not allowed.

很遗憾我们永远无法与爱因斯坦进行这样的对话，让他吐露他的科学秘密。创造力，他会同意回答我们的问题并详细解释他是如何做到的。

It’s too bad we never could have had that conversation with Einstein, the one in which he would have spilled all the secrets of his creativity, the one in which he would have agreed to answer our questions and explained in detail how he really did it.

至于格罗滕迪克，他写了一本上千页的书来讨论这个问题。他详细描述了自己做数学时脑子里在想什么。他承认，如果自己不能在脑子里构建正确的图像，他就完全无法阅读任何数学书，即使是最简单的数学书。他还承认，自己无法跟上会议的进度，因为会议对他来说总是太快。他解释了自己如何应付这种一无所知的感觉。最重要的是，他解释了自己从这一切中找到乐趣的确切地方。

As for Grothendieck, he wrote a thousand-page book on the subject. He described in detail what went on in his head when he did mathematics. He acknowledged his total inability to read any math book, even the most simple, if he didn’t manage to fabricate the right pictures in his head. He also acknowledged his inability to follow along during conferences because they always went too fast for him. He explained his way of getting by with the feeling of not knowing anything. And most of all he explained the exact place where he was finding pleasure in all that.

这个非凡的故事名为《收获与播种》（Récoltes et semailles）。手稿长期未出版，秘密流传了三十五年，直到有人终于敢于出版它。第一个合法版本出现在 2022 年，由巴黎的 Éditions Gallimard 出版。麻省理工学院出版社正在准备即将出版的英文版。

This extraordinary tale is called Harvests and Sowings (Récoltes et semailles). The manuscript remained unpublished for a long time, circulating clandestinely for over thirty-five years until someone finally dared to publish it. The first legal edition appeared in 2022, published by Éditions Gallimard in Paris. A forthcoming English-language edition is being prepared by MIT Press.

最令人叹为观止的账户

The Most Breathtaking Account

“驱动和主导我工作的，是其灵魂和存在理由，是在工作过程中为理解数学事物的现实而形成的心理意象。……我一生都无法阅读数学文本，无论它多么琐碎或简单，除非我能根据我对数学事物的经验赋予该文本以‘意义’，也就是说，除非文本在我心中唤起心理意象，即赋予它生命的直觉。”

“What drives and dominates my work, its soul and reason for being, are the mental images formed during the course of the work to apprehend the reality of mathematical things. . . . All my life I’ve been unable to read a mathematical text, however trivial or simple it may be, unless I’m able to give this text a ‘meaning’ in terms of my experience of mathematical things, that is unless the text arouses in me mental images, intuitions that will give it life.”

《收获与播种》是我读过的最迷人的书之一，书中有许多这样的启发性摘录。然而，我真的不能推荐它，或者至少不能不提出一个严重的警告：格罗滕迪克的独白是一场漫长而令人不安的咆哮，时而精彩，时而混乱，带有预言的意味，环环相扣隐喻和寓言、注释和题外话散布在页面底部，所有这些都带有自己的注释和题外话。他沉迷于数百页的个人不满和毫无根据的指责，坦率地说，几乎无法阅读。

Many such illuminating excerpts can be isolated from Harvests and Sowings, one of the most fascinating books I’ve ever read. Yet I can’t really recommend it, or at least not without a serious caveat: Grothendieck’s monologue is a long and disconcerting rant, alternatively brilliant and confused, with prophetic overtones, interlocking metaphors and allegories, notes and digressions strewn about the bottom of the page, all with their own notes and digressions. He gets lost for hundreds of pages in personal grievances and unfounded recriminations that are, quite frankly, almost unreadable.

这是一本为初学者写的书，即使是初学者也很难读完。然而，人们普遍认为《收获与播种》是有史以来最令人叹为观止的数学体验记述。

It’s a text meant for the initiated, and even the initiated have a hard time making it through to the end. A frequent opinion, however, is that Harvests and Sowings is the most breathtaking account ever written about the mathematical experience.

像我的许多数学家朋友一样，我发现了令人震惊的真理和清晰的段落，那一刻我停下阅读并对自己说：“他说得对。就是这样。这就是秘密。这就是我们头脑中真正发生的事情。正是通过这些看似无辜但却没人梦想过的简单心理活动，你才能真正擅长数学。我从未读过如此重要的书。我需要找到一种方法来告诉世界并解释格罗滕迪克想要表达的意思。”

Like many of my mathematician friends, I’ve found stunning passages of truth and clarity, moments when I stopped reading and said to myself: “He’s right. That’s it exactly. That’s really it, the secret. That’s how it really happens in our heads. It’s precisely in making these simple mental actions, which seem so innocent but which no one had dreamed of doing, that you get really good at math. I’ve never read anything as important. I need to find a way to tell the world and explain what Grothendieck is trying to say.”

然而，我知道，格罗滕迪克的思想本质上太过神秘。归根结底，这有点像爱因斯坦的问题。不再可能与他进行坦诚直接的对话，不再可能问出简单而幼稚的问题。

I’m aware, however, that in the raw, Grothendieck’s thought is far too enigmatic. In the end, it’s a bit like the problem with Einstein. There’s no longer the possibility of having a frank and direct conversation with him, to ask the simple and naïve questions.

格罗滕迪克走得比爱因斯坦远得多，给我们留下了令人难以置信的细节。但他是作为一个孤独的人，超越了他的时代，意识到人们还没有准备好听他的信息。为了理解他的叙述，我们需要将其与我们的共同经验联系起来。

Grothendieck went considerably further than Einstein and left us incredible details. But he did so as an isolated man, ahead of his time, aware that people were not yet ready to hear his message. To make sense of his account, we need to relate it to our common experience.

在与大家分享我个人从阅读中获得的、与我的经历相呼应的内容之前，我应该多介绍一下格罗滕迪克的生活和他不同寻常的个性。

Before sharing with you what I’ve personally taken away from my reading, what has echoed my own experience, I should tell you a bit more about Grothendieck’s life and his uncommon personality.

野孩子

A Wild Child

亚历山大·格罗滕迪克 1928 年出生于柏林。他的父母是被迫逃离纳粹政权的激进无政府主义者。1933 年，当他五岁时，他的母亲将他托付给汉堡路德教牧师威廉·海多恩 (Wilhelm Heydorn) 一家照顾。

Alexander Grothendieck was born in Berlin in 1928. His parents were militant anarchists who had to flee the Nazi regime. In 1933, when he was five years old, his mother entrusted him to the care of the family of Wilhelm Heydorn, a Lutheran pastor in Hamburg.

直到那时，格罗滕迪克似乎接受了相当不寻常的教育，这种教育是由他父母的无政府主义原则形成的。他的养母达格玛·海顿 (Dagmar Heydorn) 形容他刚出生时是个野孩子，脏兮兮的，毫无节制。当她把儿子托付给海顿夫妇时，汉卡·格罗滕迪克的条件是他们永远不送他去上学，永远不剪他的头发。

Up until this time, Grothendieck seems to have received a rather unusual education, formed by the anarchist principles of his parents. His adoptive mother, Dagmar Heydorn, described him upon his arrival as a wild child, dirty and lacking any form of inhibition. When she entrusted the Heydorns with her son, Hanka Grothendieck did so on the condition that they never send him to school, and never cut his hair.

海顿夫妇剪掉他的头发，送他去上学。这大概是他一生中唯一平静而“正常”的时期。他一生都与养父母保持着深厚的感情。

The Heydorns cut his hair and sent him to school. It was probably the only peaceful and “normal” period of his life. He retained throughout his life strong ties of affection with his adoptive family.

1939 年 4 月，海多恩夫妇担心格罗滕迪克的安全（格罗滕迪克的父亲是犹太人），将他送上了前往巴黎的火车，在那里他与流亡的父母团聚。不久之后，他的父亲被捕，并于 1942 年死于奥斯维辛。从 1940 年开始，格罗滕迪克和他的母亲住在法国南部的难民营里。

In April 1939, fearing for his safety (Grothendieck’s father was Jewish), the Heydorns put him on a train to Paris, where he reunited with his exiled parents. His father was arrested shortly afterwards, and would die in Auschwitz in 1942. Beginning in 1940, Grothendieck and his mother lived in the South of France, in refugee camps.

他们的生活开始变得像好莱坞电影里的陈词滥调：战后法国的难民母子，靠做家务或收割庄稼勉强维持生计；儿子的惊人天赋在孤独中形成，无人注意。

Their life began to resemble clichés from Hollywood movies: the refugee mother and son in postwar France, scratching out a living doing housework or harvesting; the son’s spectacular talent forming in isolation, without anyone taking notice.

1948 年，这位非典型学生终于被蒙彼利埃大学的一位教授发现，这位教授为他写了一封推荐信，推荐给著名数学家、影响深远的埃利·嘉当。就这样，20 岁的格罗滕迪克来到了巴黎，在当时最杰出的几位思想家的介绍下，接触到了最前沿的数学研究。

In 1948, the atypical student was finally spotted by one of his professors at the University of Montpellier, who wrote him a letter of recommendation to Élie Cartan, a prominent mathematician with great influence. This is how at twenty years old Grothendieck made his way to Paris and was introduced to cutting-edge mathematical research by some of the most brilliant minds of his time.

即将获得菲尔兹奖的劳伦特·施瓦茨让格罗滕迪克阅读他的最新论文，论文最后列出了他无法解决的 14 个问题。这是一个雄心勃勃的学生可以挖掘的优秀博士论文主题：选择一个问题，花三年时间思考它，让你的导师帮你找到一个不完整的解决方案，然后每个人都会很高兴。格罗滕迪克回房间了，几个月后才回来。他已经解决了所有 14 个问题。

Laurent Schwartz, who himself was about to receive the Fields Medal, had him read his latest article, which ended with a list of fourteen problems that he had been unable to solve. It’s the kind of list an ambitious student could dig through for a good PhD subject: choose a problem, spend three years thinking about it, get your advisor to help you find an incomplete solution, and everyone’s happy. Grothendieck went off to his room and came back a few months later. He’d solved all fourteen problems.

直到 1970 年，格罗滕迪克在将一个无名流放者与全球科学界的巅峰隔开的阶梯上不断攀升。他成为了最伟大、最强大、最优秀的人。他比任何人都努力。围绕他建立了一个完整的研究机构。他于 1966 年获得菲尔兹奖，但与他取得的其他成就相比，这仅仅是轶事。格罗滕迪克和他的学生开始了一项艰巨而富有远见的任务，从头开始重建代数几何。他们的工作仍然为当前大部分数学研究提供了基础。

Until 1970, Grothendieck climbed ever higher on the ladder that separates an unknown refugee from the heights of global science. He became the greatest, the strongest, the best. He worked harder than anyone else. A whole research institute was created around him. He won the Fields Medal in 1966, but that is merely anecdotal in comparison to everything else he accomplished. Grothendieck and his students set out on the immense and visionary task of reconstructing algebraic geometry from the ground up. Their work still provides the basis for a large part of current mathematical research.

但 1970 年，42 岁的格罗滕迪克突然中断了科学生涯。他辞去了自己创办的研究所的工作，开启了人生的新篇章，投身于激进主义和激进生态学。

But in 1970, at the age of forty-two, Grothendieck made a sudden break from his scientific career. He resigned from the institute created around him and opened a new chapter in his life, dedicated to militantism and radical ecology.

大约在 1980 年代中期，也就是这次中断后的十五年，他写了《收获与播种》。他打算写一本面向大众的书，因为他相信自己有一个重要的信息要传达。在 2010 年写的一封信中，他承认自己并没有完全成功：“这本关于我作为数学家的人生的‘反思与证言’，我承认它很难读，但对我来说意义重大，即使对其他人来说也是如此！”

It was in the mid-1980s, about fifteen years after this break, that he wrote Harvests and Sowings. His intention was to write a book for the general public because he believed he had an important message to deliver. In a letter he wrote in 2010, he acknowledged that he didn’t entirely succeed: “This ‘Reflection and Testimonial’ on my life as a mathematician, unreadable as it is I admit, has much meaning for me, if not to anyone else!”

从 1991 年到 2014 年去世，格罗滕迪克一直隐居在世间。他隐居在法国南部比利牛斯山脚下的小村庄拉塞尔，在那里他练习冥想，过着极度孤独和苦行的生活。他甚至试图完全靠蒲公英汤生活。

From 1991 until his death in 2014, Grothendieck retired from the world. He lived as a recluse in the small village of Lasserre in Southern France at the foot of the Pyrenees Mountains, where he practiced meditation and led an existence of extreme solitude and asceticism. He went as far as trying to live entirely on dandelion soup.

格罗滕迪克从未停止写作。他留下了大量数学、哲学和神秘主义的著作，其中似乎有一篇长达三万页的关于“邪恶问题”的沉思录。

Grothendieck never stopped writing. He left behind him immense quantities of mathematical, philosophical, and mystical writings, among which, it seems, is a thirty-thousand-page meditation on “the problem of Evil.”

数学经验与疯狂之间的联系似乎是我们无法忽视的一个话题。我们将在第 17 章中回顾这个问题。

The connection of the mathematical experience with madness is a subject we can’t seem to ignore. We’ll come back to it in chapter 17.

“孤独的礼物”

“The gift of solitude”

“探索是孩子们的特权。我说的是年幼的孩子，那些还不害怕犯错、不害怕看起来像傻瓜、不严肃、不随波逐流的孩子。他们也不害怕他们所看到的东西与对他们的期望、他们应该成为的样子不同，这会带来不好的品味。”

“Discovery is the privilege of the child. I’m talking about young children, children who aren’t yet afraid to make mistakes, to look like fools, not to be serious, not to act like everyone else. They’re also not afraid if the things they’re looking at have the bad taste of being different from what was expected from them, what they were supposed to be.”

这句出自《收获与播种》的话听起来像是我们以前听过千百遍的话，但这显然不是事实。即使这是事实，那对我们有什么好处呢？我们再也不会是小孩子了。

This quotation from Harvests and Sowings sounds like something we’ve heard thousands of times before, but that is clearly not true. And even if it were true, what good does that do us? We’ll never be young children again.

但这显然是个比喻。格罗滕迪克指的是“我们内心”的那个孩子，我们“已经失去了联系”。他的书实际上不是写给我们的，而是写给我们内心迷失的孩子，正如他从一开始就明确指出的那样：“我希望和你内心那个知道如何独处的人说话，和那个孩子说话，而不是和其他人说话。”

But it’s obviously a metaphor. Grothendieck is alluding to the child who is present “within us” and with whom “we have lost contact.” His book is actually addressed not to us but to the lost child within us, as he makes perfectly clear from the onset: “It’s to the one within you who knows how to be alone, to the child, that I wish to speak, and to no one else.”

格罗滕迪克通过与内心的孩子保持的亲近来解释他非凡的创造力：“在我内心深处，由于我还没有梦想过去探索的原因，某种纯真依然存在。”

Grothendieck explains his uncommon creativity by the proximity he maintains with his inner child: “In me, and for reasons I have not yet dreamed of exploring, a certain innocence has survived.”

他将这描述为“孤独的礼物”，即发现自己“独自一人聆听事物，全神贯注于孩子的游戏”的能力。

He describes this as a “gift of solitude,” the capacity to find himself “alone and listening to things, intensely absorbed in a child’s game.”

“寻找和发现，也就是提问和倾听，是世界上最简单、最自然的事情，没有人拥有独有的权利。这是我们所有人在摇篮里得到的‘礼物’。”

“Seeking and finding, that is to say, questioning and listening, is the simplest, the most spontaneous thing in the world, that no one has sole rights to. It’s a ‘gift’ that we all received in the cradle.”

无论人们如何看待格罗滕迪克，他的古怪、怪异、怪异的执着，他的话都值得一听。无论他是否用词恰当，他都很清楚自己在说什么。

Whatever one thinks of Grothendieck, of his strangeness, his eccentricities, his bizarre obsessions, it’s worth listening to him. Whether or not he’s using the right words, he clearly knows what he’s talking about.

《收获与播种》读起来就像一本瑜伽手册，从某种意义上来说，它确实如此。在隐喻和个人轶事的背后，文本描述了一种保持身体的方式、一种特殊的身体姿势、一种与语言和真理的不同寻常的关系。

Harvests and Sowings often reads like a yoga manual, and in a way that’s exactly what it is. Behind the metaphors and personal anecdotes, the text describes a certain way of holding your body, a peculiar physical attitude, an unusual relationship to language and truth.

格罗滕迪克是一位伟大的瑜伽修行者，他发明了自己的冥想技巧。该技巧的核心是一种激进的好奇心和对判断的漠然态度，我们或许可以称之为儿童姿势。

Grothendieck was a great yogi who invented his own meditation technique. It’s centered on a radical form of curiosity and indifference to judgment, what we might call the child’s pose.

所有数学家都开发了这种技术，但他们很少意识到这一点，也很少知道如何解释它。格罗滕迪克给了我们使用手册。

All mathematicians develop techniques of this sort, but they’re rarely aware of it, and rarely know how to explain it. Grothendieck hands us the user’s manual.

这种精神姿态显然是他工作方法的核心。大致来说，它包括以下内容。

This psychic posture is clearly at the heart of his work method. Roughly, here’s what it consists of.

“我或多或少相信我的断言”

“I believe, more or less, in my assertions”

打开一本你一无所知的数学书，感觉就像坐在商用喷气式飞机的驾驶座上或核能发电机的指挥所里。有很多按钮和屏幕，但你不知道它们是如何工作的，而且非常不想犯错。你很想知道这一切是如何运作的，但你不知道。正常的反应是坐在那里什么也不碰。在采取任何行动之前，你需要学习和思考。

Opening a math book on a subject you know nothing about is a bit like finding yourself in the pilot’s seat of a commercial jet or the command post of a nuclear generator. There are a lot of buttons and screens, but you have no clue how they work and an intense desire not to make a mistake. You would love to know how it all works, but you don’t. The normal reaction is to stay seated and not touch anything. Before making any move you need to study and think about it.

但如果你让任何两岁的孩子坐在驾驶座上，他们的行为就会有所不同。他们会按所有按钮，从红色或闪烁的按钮开始。

But if you put any two-year-old in the pilot’s seat, they’ll act differently. They’ll push all the buttons, starting with ones that are red or blinking.

格罗滕迪克的建议是像两岁的孩子一样。当他想理解某件事时，他会像孩子一样毫不犹豫地直接去做。他不会等到理解之后才开始做。他不假思索地行动，有点随意：

Grothendieck’s recommendation is to act like the two-year-old. When he wants to understand something, he goes straight at it, without hesitations, as a child would. He doesn’t wait to understand before launching into it. He acts without thinking, a bit haphazardly:

当我对某件事感到好奇时，无论是数学还是其他方面，我都会对其进行探究。我会探究它，而不会担心我的问题是否愚蠢，当然也不会考虑周全。问题通常以断言的形式出现——实际上，断言是一种探索性探索。我或多或少相信我的断言……通常，尤其是在研究开始时，断言是完全错误的——但为了说服自己，我还是必须做出这样的断言。

When I’m curious about a thing, mathematical or otherwise, I interrogate it. I interrogate it, without worrying about whether my question is or will seem to be stupid, certainly without it being well thought out. Often the question takes the form of an assertion—an assertion which, in truth, is an exploratory probe. I believe, more or less, in my assertions. . . . Often, especially at the outset of my research, the assertion is completely false—still, it was necessary to make it to convince myself.

我们需要明确他所说的“质询、提问和探索”是什么意思。格罗滕迪克在他的整本书中都将数学工作描述为一系列具体的物理活动。但“质询”到底是什么意思？如果我想质询事物，我该怎么做？仔细看，这一点根本不清楚。格罗滕迪克的另一句名言也是如此：倾听事物的声音。这到底是什么意思？

We need to clarify what he means by interrogating things, asking questions, and probing. Throughout his book, Grothendieck describes mathematical work as a succession of concrete physical activities. But what does interrogating things mean exactly? If I want to interrogate things, how do I go about it? Looked at closely, this isn’t clear at all. The same can be said about this other favorite phrase of Grothendieck: listening to the voice of things. What does that even mean?

只要我们在这里，什么是数学的东西？我们在哪里可以找到这些东西，我们如何与它们建立联系？

As long as we’re here, what is a mathematical thing? Where can we find these things, and how do we initiate contact with them?

格罗滕迪克从来没有费心去准确解释，无疑是因为他太习惯于谈论这些事情，以至于他忘记了他自己也必须学会如何去做。

Grothendieck never bothers to explain precisely, undoubtedly because he is so used to talking with these things that he’s forgotten that he himself had to learn how to do it.

数学事物是非数学家称之为数学概念或数学抽象的事物。它们可能由数字、集合、空间、不同类型的几何形状或其他类型的抽象结构组成。数学家更喜欢称它们为数学对象，因为将这些事物想象成可以触摸的物质对象会更容易理解它们。

Mathematical things are the things that nonmathematicians call mathematical concepts or mathematical abstractions. They may consist of numbers, sets, spaces, different kinds of geometric shapes, or other types of abstract structures. Mathematicians prefer to call them mathematical objects, because imagining these things as material objects that one can touch makes it easier to understand them.

询问事物、倾听事物的声音，意味着试图想象它们，检查你内心形成的心理图像，寻求巩固这些图像并使它们更清晰，努力揭示越来越多的细节，就像你试图回忆梦境时一样。

Interrogating things, listening to the voice of things, means trying to imagine them, examining the mental images that form within you, seeking to solidify these images and make them clearer, working at unveiling more and more details, as when you try to recall a dream.

被证明错误的乐趣

The Pleasure of Being Proved Wrong

这种方法需要用具体的术语来翻译。《收获与播种》的语言非常形象化，你可能会认为保持模糊是一种刻意的选择。

This approach needs to be translated in concrete terms. The language of Harvests and Sowings is so imagistic that you might think keeping it vague was a deliberate choice.

这种印象是错误的。格罗滕迪克力求准确。他那些神秘的词汇是为了解决一个实际问题：他描述了我们在头脑中进行的动作和我们操纵的心理图像，但我们的语言缺少合适的词汇。没有特定的词汇可以清楚地描述这些动作和图像。没有人花时间告诉我们我们有权谈论它们。

This impression is false. Grothendieck endeavored to be precise. His enigmatic vocabulary was meant to solve a practical problem: he’s describing actions that we perform in our heads and mental images that we manipulate, but our language is missing the right words. There is no specific vocabulary to talk plainly of these actions and images. No one has taken the time to even tell us we have the right to talk about them.

孩子的姿势并不是寓言，而是一种非常精确的心理态度。

The child’s pose isn’t an allegory. It’s a very precise mental attitude.

基本原理简单却具有革命性。这种想法几乎没人会想到，因为它太简单了，而且违背我们的本能。正是这种想法有可能改变一切，改变数学学习的所有层次，包括完全的初学者和自认数学不好的人。

The basic principle is simple yet revolutionary. It’s the kind of idea that almost no one thinks of because it’s too simple and it goes against our instincts. The kind of idea, precisely, that has the potential to change everything, at all levels of math learning, including the absolute beginners and the self-professed lousy at math.

当你遇到一个新的数学概念时，很难想象它。它通过抽象的定义、一串纸上的单词或教授说的话呈现给你。这串单词对你来说毫无意义。它没有直观的含义。

When you come across a new mathematical concept, it’s hard to imagine it. It is presented to you by means of an abstract definition, a string of words on a page or words spoken by a professor. This string of words doesn’t make any sense to you. It has no intuitive meaning.

学生通常不觉得自己有权想象他们还不理解的数学对象。他们觉得在敢于想象之前，他们需要了解更多。与此同时，他们满足于尝试逐字逐句地解读符号。他们可能不理解他们所读的内容，这可能会让他们头疼，但他们告诉自己，如果他们继续尝试，他们就会达到足够自信的地步，最终想象出这些文字背后的东西。但这种方法几乎从来都行不通。

Students generally don’t feel like they have the right to imagine mathematical objects that they don’t yet understand. They feel they need to know more before daring to picture them. In the meantime, they’re content to try to decipher word by word, symbol by symbol. They may not understand what they’re reading, it might give them a headache, but they tell themselves that if they keep on trying they’ll get to the point where they feel confident enough to finally imagine what’s behind the words. But this approach almost never works.

格罗滕迪克的做法不同。他知道收集尚无法看到的东西的信息毫无意义。相反，他允许自己立即想象这些事情，而不用等待，即使他很清楚这样做可能行不通，而且他脑海中的想象可能会大错特错。

Grothendieck did it differently. He knew that it was worthless to gather information about things that you can’t yet see. Instead, he allowed himself to imagine the things right away, without waiting, even when he was well aware that it might not work and his mental images would likely be terribly wrong.

他不害怕失败。他甚至确信自己会错，而这正是他所追求的。

He wasn’t afraid of failure. He was even certain that he would be wrong, and that’s exactly what he was looking for.

格罗滕迪克积极寻找错误，就像小孩子积极寻找恶作剧一样。在探索数学世界的过程中，每当他发现一些奇怪或有趣的东西、不清楚或不令人满意的东西、不连贯或不愉快的东西时，他就会开始挖掘。

Grothendieck actively sought out the error as a young child actively seeks mischief. In his exploration of the world of mathematics, each time he found something bizarre or intriguing, unclear or unsatisfactory, incoherent or disagreeable, that’s where he began digging.

当他对世界的认知出现偏差时，他就会感到不安。他四处寻找这种不安的根源，因为这是缓解这种不安的唯一方法。发现错误给他带来了快乐和解脱。“发现错误是一个关键时刻，最重要的是创造在所有发现工作中，无论是数学还是自我发现，都是一个创新时刻。这是一个我们对所研究事物的认识突然更新的时刻。”

When something was off in his vision of the world, it made him feel uneasy. He dug around to find the source of this unease, since that was the only way to relieve it. Finding mistakes gave him pleasure, relief. “Finding mistakes is a crucial moment, above all a creative moment, in all work of discovery, whether it’s in mathematics or within oneself. It’s a moment when our knowledge of the thing being examined is suddenly renewed.”

格罗滕迪克关于错误的论述具有普遍意义，远远超出了科学领域。它让你想把他的话刻在学校的门面上：

What Grothendieck wrote about error is of universal significance, well beyond the field of science. It makes you want to engrave his words on school façades:

害怕犯错和害怕真相是一回事。害怕犯错的人无力发现任何新东西。当我们害怕犯错时，我们内心的错误就会变得坚如磐​​石。

Fear of mistakes and fear of the truth is one and the same thing. The person who fears being wrong is powerless to discover anything new. It’s when we fear making a mistake that the error which is inside of us becomes immovable as a rock.

很少有人意识到，数学学习的主要障碍是心理上的，不仅是在开始阶段，而且在整个学习过程中，直到最高水平。当我们告别童年时，我们学会了害怕显得愚蠢。我们学会了为自己的错误感到羞耻。我们学会了隐藏我们几乎一无所知的事实，甚至对自己也是如此。要想在数学上取得进步，我们需要停止这种掩饰的习惯。这并不容易。

Not many people realize that the main obstacles in mathematics are psychological, not only at the beginning but all throughout, up to the highest levels. As we leave childhood behind, we learn to fear looking stupid. We learn to be ashamed of our mistakes. We learn to hide, even to ourselves, the fact that we know almost nothing. To get ahead in math, we need to deactivate this reflex for dissimulation. And it’s not easy.

在我们还可以自由地问愚蠢的问题，甚至连续问几百次相同的愚蠢问题的年龄，没有人讨厌数学。伟大的数学家发明并实施了特殊的技术来恢复这种失去的童年纯真。他们都说这是不可或缺的。我们将在第 13 章中回顾这一点。

At the age when we were still free to ask stupid questions, even to ask the same stupid questions hundreds of times in a row, no one hated math. The great mathematicians invent and put in place special techniques to recover this lost childhood innocence. They all say it’s indispensable. We’ll come back to this in chapter 13.

学习的驱动力

The Driving Force of Learning

当格罗滕迪克谈到“我们内心的错误”时，这与逻辑无关。这不是计算或推理的错误。格罗滕迪克所说的错误是直觉错误，是视觉错误：我们对事物的印象并不正确。

When Grothendieck talks of “the error which is inside of us,” that has nothing to do with logic. It’s not an error of computation or reasoning. The error Grothendieck is talking of is one of intuition, an error of vision: the image that we have of things isn’t correct.

正如我们将在本书中看到的，数学理解是通过逐步改变我们向自己呈现事物的方式来实现的，使其更清晰，更精确，更接近现实。

As we’ll see throughout this book, mathematical understanding is achieved by gradually modifying the way we represent things to ourselves, and making them clearer, more precise, closer to reality.

有时你会听到人们说，我们的大脑左右半球功能不同。左脑擅长逻辑推理和计算，右脑则擅长联想推理和直觉。

You sometimes hear people say that the two hemispheres of our brains function differently. The left side of the brain specializes in logical reasoning and calculation, while the right side specializes in associative reasoning and intuition.

这种对我们解剖学的荒谬解释可以追溯到 20 世纪 60 年代，并且早已被推翻。实际上，我们大脑的两侧非常相似，从深层次上讲，两者都具有联想和直觉功能。让你以逻辑方式看待世界的器官并不存在。如果你指望它能让你擅长数学，那你就得等很长时间了。

This nonsensical interpretation of our anatomy dates from the 1960s and has long since been discredited. In reality the two sides of our brains closely resemble one another and, at a profound level, both function associatively and intuitively. The organ that allows you to see the world in a logical manner doesn’t exist. If you’re counting on that to become good at math, you’ll have a long wait.

我们非凡的学习和发明能力源于我们无意识的能力，即不断重新配置图像和感觉的联想结构，从字面上和比喻上讲，这些结构构成了我们思维的真实结构。

Our prodigious faculty for learning and invention has its origin in our unconscious ability to constantly reconfigure the fabric of associations of images and sensations that, literally and figuratively, comprise the real structure of our thought.

我们所有伟大的学习成就都依赖于这种心理可塑性。错误起着根本性的作用，因为它是可塑性的驱动力。学习看东西、走路、使用勺子、系鞋带、说话、阅读和写作，总是需要重新配置你的大脑。而且这绝非一蹴而就。孩子们只有在尝试和失败之后才会学会走路。他们需要跌倒才能学会站起来。正是错误的积累让他们发展出了直觉的平衡感。

All our great learning achievements rely on this mental plasticity. Error plays a fundamental role, as it is the driving force of plasticity. Learning to see, to walk, use a spoon, tie your shoelaces, talk, read and write, is always about reconfiguring your brain. And it’s never done in one shot. Children don’t learn how to walk until they’ve tried and failed. They need to fall in order to learn how to stand up. It’s the accumulation of errors that allow them to develop their intuitive sense of balance.

就像每个运动学习一样，理解一个新的数学概念需要通过重新配置直觉来进行，而这需要一个“试探”阶段。一旦转移到行走的背景下，格罗滕迪克对错误作用的评论就变得更加具有启发性：

As for every motor learning, understanding a new mathematical concept proceeds by a reconfiguring of intuition, and that requires a “feeling-out” phase. Once transposed to the context of walking, Grothendieck’s comments on the role of error become even more illuminating:

害怕跌倒和害怕走路其实是同一件事。害怕摔倒的人是没有能力学会走路的。 只有当我们坐在地上时，我们最初的笨拙才会变成身体上的残疾。

Fear of falling and fear of walking are one and the same thing. The person who fears falling on their face is powerless to learn how to walk. It’s only when we stay on our ass that our initial clumsiness turns into physical disability.

逻辑的作用

The Role of Logic

在心理意象的世界里，物理定律不适用。你可以想象任何事情，甚至是不一致的事情，而不会让你失望。我们内心的错误可能会变得像石头一样不可动摇，而我们甚至没有意识到。

In the world of mental images, the laws of physics don’t apply. You can imagine anything, even inconsistent things, without falling on your face. The error that is inside of us can become immovable as a rock without our even being aware of it.

正是在这里，数学方法与我们通常使用直觉的方式不同。数学家发明了一种让他们发现自己内部错误的方法。这种方法依赖于写作——更准确地说，依赖于以逻辑形式主义为基础的数学官方语言的写作。

It’s precisely here that the mathematical approach diverges from our usual way of using our intuition. Mathematicians have invented a method that lets them discover the errors inside themselves. This method relies on writing—more precisely, on writing in the official language of mathematics, constructed on logical formalism.

逻辑并不能帮助你思考。它只能帮助你找出你的错误思维。

Logic doesn’t help you think. It helps you find out where you’re thinking wrong.

当格罗滕迪克发出“探测器”去询问他想要了解的对象时，他通过以下写作得到答案：

When Grothendieck sends out “probes” to interrogate objects he wants to understand, he gets his answer by writing:

通常，你只需要写下来就能发现它不正确，而在写下之前，你会感到模糊，感觉不好，而不是有证据。现在，你可以重新开始，不再缺乏知识，提出一个问题断言，也许问题会少一些。

Often, you only have to write it down for you to see it’s incorrect, whereas before writing there was a vagueness, a bad feeling, instead of this evidence. That now allows you to start over without this lack of knowledge, with a question-assertion perhaps a little less off the mark.

与生物学家在完成实验后才撰写论文不同，数学家在研究工作期间写作，因为写作本身就是研究的一部分。格罗滕迪克说：

As opposed to biologists, who write their articles only after having done their experiments, mathematicians write during their research work, because writing is itself part of the research. Here’s what Grothendieck says:

写作的作用不在于记录研究的结果，而在于研究的过程本身。

The role of writing is not to record the results of research, but is the process itself of research.

我一直竭尽全力，用数学语言尽可能细致地描述这些图像及其带来的理解。正是在这种不断努力表达无法表达的东西、定义尚不清楚的东西的过程中，数学工作（或许也是所有创造性智力工作）的特殊动力或许得以发现。

I have always made every effort to describe in the most meticulous way possible, using mathematical language, these images and the understanding they bring. It is in this continuous effort to articulate the inarticulable, to define what is as yet unclear, that the particular dynamic of mathematical work (and perhaps as well all creative intellectual work) is perhaps found.

数学写作是将活生生的（但混乱、不稳定、非语言的）直觉转录成精确而稳定的（但像化石一样死气沉沉的）文本的工作。

Mathematical writing is the work of transcribing a living (but confused, unstable, nonverbal) intuition into a precise and stable (but as dead as a fossil) text.

或者说，如果直觉从一开始就是精确和正确的，那么这将是一项简单的抄写工作。但直觉很少从一开始就是精确和正确的。起初它是模糊和错误的，而且总是有点模糊和错误。通过写作工作，直觉变得越来越不模糊，越来越不错误。这个过程是缓慢而渐进的。

Or, rather, it would be a simple job of transcription if the intuition was from the outset precise and correct. But intuition is rarely precise and correct from the outset. At first it’s vague and wrong, and it always remains a bit so. Through the work of writing, intuition becomes less and less vague and less and less wrong. This process is slow and gradual.

数学创造是想象力（想象你所读到的东西的艺术）和语言表达（用语言表达你所看到的东西的艺术）之间的不断反复。这同时改变了我们的直觉和语言。我们学会了看，同时也学会了说话。我们学会了想象新事物，并发明了一种可以让我们命名它们的语言。根据格罗滕迪克的说法，整个过程相当于“从表面上的虚空中聚集无形的雾气”。

Mathematical creation is a constant back-and-forth between imagination (the art of picturing what you read) and verbalization (the art of putting words to what you see). This simultaneously transforms our intuition and our language. We learn to see and, at the same time, we learn to talk. We learn to picture new things and we invent a language that allows us to name them. The whole process, according to Grothendieck, amounts to “gathering intangible mists from out of an apparent void.”

这项工作的成果以两种不同的方式呈现。第一种化身是看不见的：它是对世界理解和创作者意识状态的修改。第二种化身是数学文本。

The result of this work shows up in two different ways. The first incarnation is invisible: it’s the modification of the understanding of the world and the state of consciousness of the person who produced the work. The second incarnation is the mathematical text.

格罗滕迪克知道，他只能展示这个第二版，也就是印刷版。但这并不是他的动力。对他来说，“在这个形式中，你无法找到理解数学事物的灵魂。”

Grothendieck knew that this second incarnation, the printed one, was the only one he could show. But it wasn’t what drove him. For him, “it’s not in this form that you find the soul of understanding mathematical things.”

写作的努力让格罗滕迪克培养了自己的直觉。一旦有了清晰的想法，他就可以客观地看待自己的文章，就像它们是烤面包机的使用手册一样。

The effort of writing allowed Grothendieck to develop his own intuition. Once he had a clear idea, he could look at his own articles with detachment, as if they were user manuals for toasters.

梁龙

The Diplodocus

在下一章中，我们将看到数学语言的特殊性如何使其成为一种令人难以置信的思维澄清工具。

In the next chapter, we’ll see how the peculiarities of mathematical language make it an incredible tool of mental clarification.

但是，我们将通过回顾本文开头的谜团来结束本章：1956 年 11 月 13 日，28 岁的年轻的格罗滕迪克写给塞尔的信，信中他以一种随意的语气宣布他刚刚完成了他的“荒谬之作”。

But we’ll end this chapter by going back to the mystery we began with: the casual tone of the letter that the young Grothendieck, at age twenty-eight, wrote to Serre on November 13, 1956, to announce that he’d just finished his “ridiculous piece.”

1955 年 6 月，也就是 17 个月前，格罗滕迪克写信给塞尔，与他分享了他的第一篇笔记。格罗滕迪克的语气很热情，因为他正处于探索的初始阶段。他探索事物，犯了大量错误，并取得了快速的进步。当时，他仍然将笔记中的一些段落称为“未下的蛋”，可能“搞砸了”。

In June 1955, seventeen months earlier, Grothendieck had written to Serre to share with him his first notes. The tone was enthusiastic, as Grothendieck was in the initial phase of discovery. He probed into things, made massive errors, and achieved rapid progress. At the time he still qualified some passages in his notes as “unlaid eggs” that were potentially “screwed up.”

在接下来的一年里，格罗滕迪克一直在思索他的“蛋”。他看着它们孵化，耐心地喂养这个奇怪的生物。随着手稿的成长和结构逐渐成型，塞尔和格罗滕迪克谈论它时越来越流利，甚至给它起了个绰号：“梁龙”。

In the year that followed, Grothendieck brooded upon his “eggs.” He watched them hatch and patiently fed the bizarre creature that emerged. As the manuscript grew and gained structure, Serre and Grothendieck spoke about it with a growing glibness, going so far as to give it a nickname: the “diplodocus.”

伟大的想法已经成型。发现的乐趣、最终理解的乐趣已经消退。惊喜越来越少。现在只是进行最后的润色、技术细节、遵守数学官方语言的官僚要求的问题。

The big ideas were in place. The pleasure of discovery, the pleasure of finally understanding, was already fading. Surprises were becoming rare. It was simply a matter of putting on the finishing touches, the technical details, of conforming to the bureaucratic requirements of the official language of mathematics.

在编辑的最后几个月里，写作变成了一件苦差事。格罗滕迪克开始担心是否有人愿意发表他这篇荒谬的文章。他选择了日本东北数学杂志，因为“他们似乎并不介意长篇文章。”

In the final months of editing, the writing became an ordeal. Grothendieck began to worry whether anyone would want to publish his ridiculous piece. He chose the Japanese Tohoku Mathematical Journal because “it seems that really long articles don’t bother them.”

在写给塞尔的信中，格罗滕迪克甚至为自己辩解。他可能创造了一个怪物，但他别无选择：“这是我唯一能通过坚持不懈来理解事物运作方式的方法。”

In his letter to Serre, Grothendieck goes as far as excusing himself. He may have created a monster, but he didn’t have any choice: “It’s the only way I have of understanding, through sheer persistence, how things work.”

8

触觉理论

8

The Theory of Touch

在人们从不阅读的书籍中，除了数学书和烤面包机的用户手册外，人们一定不能忘记字典。

Among the books one never reads, apart from math books and user manuals for toasters, one mustn’t forget dictionaries.

当我还是个孩子的时候，我对字典很着迷。它们承诺用其他单词来定义每个单词。但它们能实现这个承诺吗？它们真的能成为语言的门户吗？当你想从头开始学习单词时，你会翻到哪一页？

When I was a child, I was fascinated by dictionaries. Their promise is to define every word using other words. But do they keep this promise? Can they really function as a gateway to language? When you want to learn words from the ground up, which page do you turn to?

如果你不知道香蕉是什么，字典会告诉你它是“香蕉树的一种细长弯曲的热带水果，这种香蕉成串生长，果肉呈奶油色，果皮光滑；特别是卡文迪什香蕉品种的甜黄色果实。”但香蕉树是什么？它是“一种长满香蕉串的热带树状植物，属于芭蕉属（但有时也包括来自恩塞特的植物），有大而细长的叶子。”

If you don’t know what a banana is, the dictionary will teach you it’s “an elongated curved tropical fruit of a banana plant, which grows in bunches and has a creamy flesh and a smooth skin; in particular, the sweet, yellow fruit of the Cavendish banana cultivar.” But what is a banana plant? It’s “the tropical tree-like plant which bears clusters of bananas, a plant of the genus Musa (but sometimes also including plants from Ensete), which has large, elongated leaves.”

这没错，但确实让人困惑。这个定义奇怪地扭曲和复杂，最重要的是，它是循环的：香蕉是香蕉植物的果实，而香蕉是水果。为什么不直截了当地说香蕉就是香蕉呢，因为这似乎是主要信息？

That’s not wrong, but it’s pretty confusing. The definition is bizarrely tortured and complicated, and most of all, it’s circular: the banana is the fruit of a banana plant, which have bananas for fruit. Why not cut to the chase and just say a banana is a banana, as this seems to be the main message?

你不会用一堆复杂的短语向一个不了解香蕉的人解释什么是香蕉。为了表明我们对香蕉的真实想法，最简单、最诚实的定义仍然是我们给孩子们的定义：“试试看！很好吃！”

You don’t explain what a banana is to someone who doesn’t know with a bunch of convoluted phrases. To show what we really think about bananas, the simplest and most honest definition is still the one we give to children: “Try it! It’s good!”

字典里充斥着循环的定义。

Dictionaries are filled with circular definitions.

什么是热？“热的状态。”什么是热？“具有或散发出高温。”什么是温度？“冷或热的量度，通常可用温度计测量。”什么是温度计？“用于测量温度的仪器。”什么是真理？“真实的特征。”真实？“符合真理的事物。”

What is heat? “The condition of being hot.” What is hot? “Having or giving off a high temperature.” What’s a temperature? “A measure of cold or heat, often measurable with a thermometer.” What’s a thermometer? “An apparatus used to measure temperature.” What is truth? “Characteristic of what is true.” True? “That which conforms to the truth.”

从逻辑上讲，字典就是庞氏骗局。如果人们真的依赖字典来了解香蕉，那么这种骗局早就被揭穿了。

From a logical standpoint, dictionaries are giant Ponzi schemes. If people truly relied on them to find out about bananas, the fraud would have been denounced long ago.

但这不是我们的做法。我们的方法不合逻辑。我们不会通过单词的定义来学习单词。我们通过不断澄清来一点一点地吸收语言。我们的大脑有能力在知道如何命名之前看到事物，在理解单词的含义之前识别单词，并逐渐将单词与我们看到的东西联系起来。

But that’s not how we do it. Our approach isn’t logical. We don’t learn words through their definitions. We assimilate language bit by bit, by successive clarifications. Our brain has the ability to see things before knowing how to name them, to recognize words before understanding their meaning, and to gradually associate the words with what we see.

我们从零开始，确实是从零开始。我们不是从字典开始。我们从生活和与他人分享的共同经历开始。

We start from zero, literally. We don’t start from dictionaries. We start with life and the common experiences we share with others.

从零开始

Starting from Zero

数学定义类似于字典中的定义，但有一个细微的区别：它们实际上定义了事物。

Mathematical definitions resemble definitions in dictionaries, with one slight difference: they actually define things.

与字典不同，数学书并不是简单地将已经存在的单词联系起来。它们并不局限于你可以指出的事物或我们有共同经验的事物。

As opposed to dictionaries, math books don’t simply make connections between words that already exist. They don’t limit themselves to things you can point to or that we have a shared experience with.

数学定义既不是评论，也不是解释：它是新思维形象的精确组装指南，也是选择用来指代它的新词的“出生证明”。（在实践中，现有的单词经常被重复使用，获得新的含义，可能与这些词在日常生活中的含义没有直接关系。

A mathematical definition is neither a commentary nor an explication: it is the exact assembly guide of a new mental image and the “birth certificate” of the new word chosen to designate it. (In practice, existing words are often reused, receiving new meanings that may have no direct relation to what these words mean in everyday life.)

从这个意义上说，数学定义具有创造的力量：它们使事物存在。这样夸张的说法可能显得很愚蠢，但事实确实如此。

In that sense, mathematical definitions have the power of creation: they bring things into existence. It may seem silly to speak so pompously, but that’s what really happens.

当你看到别人尚未察觉的事物时，分享你的愿景需要找到一种方法让别人在自己的头脑中重新创造这些事物。数学定义就是为此目的而生的。它提供了详细的说明，让其他人从他们已经能够看到的事物开始，在头脑中构建这些新事物。

When you see things that others don’t yet perceive, sharing your vision requires finding a way to get others to re-create those things in their own heads. A mathematical definition serves this purpose. It provides detailed instructions allowing others, starting with things they are already able to see, to mentally construct those new things.

巨大的扩张因素

An Enormous Factor of Expansion

理论上，每个人都应该能够阅读数学书籍。与字典不同，它们不包含循环定义。不需要隐性知识，并且当需要时，读者可以参考以前的参考资料，在那里他们可以找到他们还不熟悉的单词的定义。由于说明很明确，并且给出了所有细节，因此理解起来应该不会有任何障碍。

In theory, everyone should be able to read math books. Unlike dictionaries, they contain no circular definitions. No implicit knowledge is required, and whenever it is necessary readers are referred back to previous references where they can find the definition of words they’re not yet familiar with. Since the instructions are clear and all the details are given, there shouldn’t be any obstacles to understanding.

然而在实践中，从一本数学书的开头我们就面临着一个巨大的问题：用语言解释心理图像极其困难。

In practice, however, from the beginning lines of a math book we’re faced with a huge problem: it’s fantastically difficult to explain a mental image in words.

正如瑟斯顿所说：“将我自己思维中的编码转化为可以传达给别人的东西，有时需要很大的扩展因素。”

As Thurston remarked, “There is sometimes a huge expansion factor in translating from the encoding in my own thinking to something that can be conveyed to someone else.”

结果往往令人难以接受。瑟斯顿说“巨大的扩展因素”时，他的意思并不是说它会长两倍或三倍。他的意思是，对于我们来说显而易见的事情，书面记录可能比我们在脑海中为自己做的总结长十倍、一百倍或一千倍。即便如此，你也会不得不抛开一堆你不忍心写下来的细节。

The result is often unpalatable. When Thurston speaks of a “huge expansion factor,” he doesn’t mean that it will be two or three times as long. He means that the written transcription of what seems obvious to us might be ten, a hundred, or a thousand times as long as the summary we make for ourselves in our head. And even then you’ll have to leave aside a bunch of details you won’t have the heart to write down.

瑟斯顿描述的现象并不局限于高级研究。当我们试图忠实地描述我们脑海中最简单的图像时，这种现象就会出现。一张图片胜过千言万语，不幸的是，这也适用于只存在于我们头脑中的图像。

The phenomenon Thurston describes isn’t limited to high-level research. It shows up as soon as we try to faithfully describe even the simplest of our mental images. An image is worth a thousand words, and unfortunately that also applies to images that exist only in our heads.

为了更好地理解这一点，让我们举一个我们最喜欢的例子。你需要多长时间才能想象出你系鞋带时做了什么？两秒？三秒？现在拿起笔和纸，试着准确描述每个动作，这样完全的初学者就可以按照你的指示得到相同的结果。这个练习有一个困难版本，只允许使用文字。但更简单的版本，你可以使用绘图，已经非常难了。

To get a better idea of this, let’s pick up one of our favorite examples. How much time does it take you to visualize what you do when you tie your shoes? Two seconds? Three? Now take pen and paper and try to describe each movement exactly, so that an absolute beginner could follow your instructions and get the same result. There is a difficult version of this exercise where only words are allowed. But the easier version, where you can use drawings, is already immensely hard.

一旦你掌握了这种难度的程度，你就会明白一些基本而又令人深感欣慰的事情：一本数学书可能看起来非常复杂，即使其中的思想很简单。

Once you grasp the degree of this difficulty, you’ll understand something fundamental and profoundly comforting: a math book might seem to be terribly complicated even though the ideas are quite simple.

真的没有理由害怕数学著作。瑟斯顿所说的字里行间读懂不仅是可能的，而且它一定比文本本身更简单。

There’s really no reason to be afraid of mathematical writings. Not only is the reading between the lines Thurston talks about possible, but it’s necessarily simpler than the text itself.

但是在获得这种简单的理解之前，当您还没有正确的心理图像时，您还必须经过大量的反复试验。

But before getting to this simple understanding, while you still don’t have the right mental images, you’ll have to go through a lot of trial and error.

清晰的艺术

The Art of Clarity

如果你找不到词语来描述你如何系鞋带，这并不奇怪。很可能你根本就没试过。写数学，也就是以足够清晰和精确的方式转录心理图像，以便让其他人理解和重现它们，是一门艺术。

It’s not surprising if you couldn’t find the words to describe how you tie your shoes. Most likely, you didn’t even try. Writing mathematics, that is, transcribing mental images with enough clarity and precision to allow others to understand and reproduce them, is an art.

之所以如此困难，是因为你的心理形象往往比你想象的要模糊得多。是什么让你打的鞋带不松开？如果你不知道，你就真的不知道如何才能把鞋带系好。

What makes it so difficult is that your mental images are often a lot less clear than you think. What keeps the knot you make in your shoelaces from coming undone? If you don’t know, you don’t really know what it takes to tie it right.

正如格罗滕迪克所解释的那样，数学写作艺术实际上是一个双重任务，既要澄清思想，又要完善语言。这是一项精细的运动协调练习，需要多年的练习才能掌握。好消息是，只要有耐心和努力，任何人都有能力变得更好。

As Grothendieck explained, the art of mathematical writing is really a dual task of the clarification of ideas and the refinement of language. It’s a delicate exercise of motor coordination and requires years of practice to master. The good news is that with patience and effort anyone has the ability to get better.

学习写数学就是学习有清晰的思路。如果你没有这个能力，那岂不是太可惜了？

Learning to write math is learning to have clear ideas. Wouldn’t it be a shame to deprive yourself of that?

通过自己编写数学程序，你就会明白为什么它会用如此奇怪的形式主义、用这种为机器人设计的语言来编写：真的没有其他选择。

By writing math yourself, you’ll get to understand why it is written in such a bizarre formalism, in this language made for robots: there’s really no other choice.

为了说明这一点，我们将回到我们最喜欢的另一个例子：形状的概念，比如你在童年时期发现的概念。想象在另一个宇宙中，你是第一个发现形状概念的人。你会如何用语言解释你区分星星和正方形并将正确的积木放入正确的洞中的方法？

To demonstrate, we’ll return to another of our favorite examples: the concept of shapes, such as you discovered in your early childhood. Imagine an alternative universe where you were the first person to discover the concept of shapes. How would you explain in words your method of distinguishing stars from squares and putting the right blocks in the right holes?

耐心游戏

The Patience Game

在这个替代宇宙中，视觉文化非常贫乏，以至于形状游戏被称为耐心游戏，因为解决问题的唯一已知方法是经过数小时的反复试验。

In this alternative universe the visual culture is so impoverished that the shape game is called the patience game, because the only known method to solve the problem is through hours of trial and error.

没有几何语言。没有“圆形”、“方形”或“三角形”的词。心脏这个词只用于指我们胸部跳动的肌肉。如果你用这个词来指出耐心游戏中的一块棋子，没有人会理解你。星星这个词也是一样。星星在夜空中闪耀，但没有人会看到它与耐心游戏的关系。问题甚至不是星星有五点、六点还是七点。人们离这个还差得很远。而且，你从哪里得到星星应该有点的想法？那到底是什么意思？

There’s no geometric language. There’s no word for “round,” “square,” or “triangle.” The word heart is used only for the muscle that beats in our chest. If you used this word to point out one of the pieces in the patience game, no one would understand you. The same thing with the word star. Stars shine in the night sky but no one would see the relation to the patience game. The question’s not even whether the stars have five, six, or seven points. People are far from that. And besides, where do you get the idea that stars should have points? What does that even mean?

在你看待世界的方式中，在你的内心语言中，你接受星星有尖角的想法，并且你认为五角星是耐心游戏中的方块之一。为什么不呢？只是到目前为止，这个想法只存在于你的脑海中。

In your way of seeing the world, in your inner language, you accept the idea that stars have points and you recognize a five-pointed star as one of the blocks in the patience game. Why not? Except that so far this idea exists only in your head.

其他人并不盲目。从生理上讲，他们能够看到和你一样的形状。但他们还没有学会如何做到这一点。他们的大脑接收相同的原始视觉信息，但不会以相同的方式构建它。

The others aren’t blind. They are biologically capable of seeing the same shapes as you. But they haven’t yet learned how. Their brain receives the same raw visual information but doesn’t structure it in the same way.

“看，这块积木是星形的。上面还有一个星形的洞。如果你把这块积木拿来，用正确的方法放进这个洞里，它第一次就能进去。”

“Look, this block is in the shape of a star. There’s a hole also in the shape of a star. If you take this block and put it in this hole, the right way, it goes in on the first try.”

这种解释行不通。人们看不到眼前的星星。他们可能和你生活在同一个世界，但他们的心理体验却不同。如果他们看到你不费吹灰之力就解决了游戏，他们会大笑起来。你看起来像个魔术师。

This kind of explanation doesn’t work. People can’t see the star that’s in front of their eyes. They may very well live in the same world as you, but their mental experiences are different. They’d laugh out loud if they saw you solve the game without trial and error. You’d look like a magician.

触觉理论

The Theory of Touch

与几何学不同，平行宇宙的居民拥有高度发达的触觉。小学生都学过触觉理论。他们学习如何用手指沿着表面滑动并识别纹理。他们学习柔软、坚硬、光滑、粗糙、凹槽、纤维状、粗糙、易碎、多孔等概念。

Instead of geometry, the inhabitants of the alternative universe have a highly developed sense of touch. Schoolchildren are all taught the theory of touch. They learn how to run their fingers along surfaces and recognize textures. They learn soft, hard, smooth, rough, grooved, stringy, raspy, friable, porous.

当他们拿着一个积木时，他们很容易就找到一些突出的边缘（他们称之为“点”），以及一些手指被卡住的凹槽（这些被称为“坑”）。

When they hold a block, they readily identify some protruding edges (what they call points) and also some concave grooves where their fingers get stuck (these are called pits).

虽然可能不多，但这是一个起点。数学的力量在于它能够用精确定义的新词来扩展语言。通过依赖人们已经理解的事物，你可以构建新的事物，这些事物他们无法立即感受到，但他们仍然可以通过定义来操纵。

It might not be much, but it’s a starting point. The power of math lies in its ability to extend language with new words that are precisely defined. By relying on things people already understand, you construct new ones that they can’t immediately feel but that they can nevertheless manipulate through definitions.

通过尝试这些新单词，希望他们最终有一天能够真正理解它们。

By playing around with these new words, hopefully they’ll end up someday really understanding them.

如果不依赖视觉语言来谈论三角形、星形和正方形，你就必须依赖触觉词汇来重构这些概念。你不能简单地用手指指着说：“看，那是一颗星星。”这种无法依赖共同经验的情况会给写作带来巨大困难，最终你会写出一篇复杂而深奥的文字。但这是可行的。

To talk about triangles, stars, and squares without relying on the language of vision, you have to reconstruct these concepts by relying on the vocabulary of touch. You can’t simply point your finger and say, “Look, that’s a star.” This inability to rely on a shared experience will pose tremendous difficulties in writing and you’ll end up with a complicated and unfathomable text. But it’s doable.

结果可能看起来是这样的。小心：接下来的几页是用一种非常类似于官方数学的风格写的。因此，坦率地说，读起来很痛苦。

This is what the result might look like. Careful: the following few pages are written in a style that greatly resembles official math. Consequently, they are quite frankly a pain to read.

耐心游戏的触觉理论（适合初学者）

Tactile Theory of the Patience Game, for Beginners

当你用手指沿着木块（或孔）的边缘滑动时，你会看到一系列的点和凹坑。让我们给这一系列的点和凹坑起个名字，称之为木块（或孔）的签名。例如，有一个块（你想称之为三角形，但这个词还不存在），其签名是：

When you run your finger along the edge of a block (or hole), you come across a series of points and pits. Let’s give a name to this series of points and pits, and call it the signature of the block (or hole). For example, there’s a block (the one you want to call triangle, except that this word doesn’t yet exist) whose signature is:

点，点，点

point, point, point

以及一个洞（适合该块的洞），其签名为：

and a hole (the one that fits the block) whose signature is:

坑，坑，坑。

pit, pit, pit.

和数学定义一样，名称是任意的。我选择了“签名”，但我也可以选择其他名称，这不会改变任何东西，因为我赋予该词的含义完全包含在定义中，与其日常含义没有任何直接关系。但既然我可以选择，我不妨选择一个对读者有帮助的词，这个词的日常含义可能有助于他们理解数学含义。签名对我来说似乎不错，因为它唤起了这样的想法：签名可以让你识别单个块和孔。如果这个词不适合您，请随意使用其他词。

As always with mathematical definitions, the name is arbitrary. I chose “signature,” but I could have chosen a different name and it wouldn’t change anything, since the meaning I assign to the word is entirely contained in the definition, without any direct relation to its everyday meaning. But since I have the choice, I may as well pick a word that helps readers, a word whose everyday meaning might help them understand the mathematical one. Signature seems good to me since it evokes the idea that the signature lets you identify individual blocks and holes. If this word doesn’t work for you, feel free to use another.

然而，我给出的定义有一个小小的技术问题：同一个对象可以有多个不同的签名，这取决于你从哪里开始。因此，为了严格起见，我们应该谈论“一个”签名，而不是“那个”签名。例如，以一个区块（你想称之为星形的区块）为例，它的签名是：

The definition I’ve given, however, has a minor technical problem: the same object can have a number of different signatures, depending on where you start. To be rigorous, therefore, we should talk about “a” signature rather than “the” signature. For example, take a block (the one you want to call a star) whose signature is:

点，坑，点，坑，点，坑，点，坑，点，坑。

point, pit, point, pit, point, pit, point, pit, point, pit.

但是如果你从不同的地方开始，你也可以得到这个签名：

But if you start your finger at a different place you could also come up with this signature:

坑，点，坑，点，坑，点，坑，点，坑，点。

pit, point, pit, point, pit, point, pit, point, pit, point.

重要的不是签名本身，而是旋转前的签名。但这个概念需要正确定义。为此，我首先需要一个中间概念：签名的基本旋转定义为取第一个字并将其放在末尾所获得的签名。例如，

What matters isn’t the signature itself but the signature up to rotation. But this notion needs to be properly defined. To do this, I first need an intermediate notion: the elementary rotation of a signature is defined as the signature obtained by taking the first word and putting it at the end. For example, the elementary rotation of

坑，点，点，点

pit, point, point, point

是

is

点，点，点，坑。

point, point, point, pit.

如果可以通过一系列基本旋转从一个签名转到另一个签名，则这两个签名在旋转之前是等效的。例如，以下四个签名在旋转之前是等效的：

The two signatures are equivalent up to rotation if you can go from one to the other by a series of elementary rotations. For example, the four following signatures are equivalent up to rotation:

坑，点，点，点

pit, point, point, point

點，點，點，坑

point, point, point, pit

點，點，坑，點

point, point, pit, point

点，坑，点，点。

point, pit, point, point.

每行都是通过应用基本旋转从前一行得到的。如果你对最后一行再应用一次基本旋转，你就会回到第一行。

Each line is obtained from the previous one by applying an elementary rotation. If you apply yet another elementary rotation to the last line, you’ll get back to the first.

定义。形状是旋转为止的签名等价类。

Definition. A shape is an equivalence class of signatures up to rotation.

要理解这个定义，你需要知道什么是等价类。这是一个基本的数学概念，你可以在任何集合论教科书中找到它的定义。在实践中，这意味着任何签名都定义一种形式，并且两个签名定义相同的形式当且仅当它们在旋转之前等价时。综合起来，上述四个签名构成了旋转之前签名等价类的一个例子。

To make sense of this definition, you need to know what an equivalence class is. It’s a basic mathematical concept, and you’ll find the definition in any set theory textbook. In practice, it means any signature defines a form and that two signatures define the same form if and only if they are equivalent up to rotation. Taken together, the four signatures above constitute an example of an equivalence class of signatures up to rotation.

如果我愿意，我可以继续创造词语。我可以根据三角形、圆形和正方形的特征来定义它们。例如，三角形可以定义为具有以下特征的形状：

If I wanted to, I could continue to invent words. I could define triangles, circles, and squares in terms of their signatures. For example, a triangle would be defined as a shape whose signature is:

点，点，点。

point, point, point.

同样，我可以选择将一颗有 n 个点的星定义为其签名是通过重复n次公式point, pit 获得的。在五角星的具体情况下，签名将是：

In the same way I could choose to define a star with n points as the form whose signature is obtained by repeating n times the formula point, pit. In the particular case of the five-pointed star, the signature would be:

点，坑，点，坑，点，坑，点，坑，点，坑。

point, pit, point, pit, point, pit, point, pit, point, pit.

心形，顾名思义，是一颗只有一个点的星。

A heart, by definition, is a star with a single point.

签名和形状的语言将使我们能够描述一种消除耐心需要的耐心游戏解决方案。

The language of signatures and shapes will enable us to describe a solution to the patience game that eliminates the need for patience.

定义。签名的镜像是通过系统地将单词pit替换为单词point，将单词point替换为单词pit ，从签名中获得的一系列单词。

Definition. The mirror image of a signature is the series of words obtained from the signature by systematically replacing the word pit with the word point and the word point with the word pit.

因此，点、点、点的镜像是坑、坑、坑，反之亦然。如果两个签名在旋转时等价，则它们的镜像也是等价的，因此镜像的概念可以扩展到形状。耐心博弈理论的主要结果如下：

Thus the mirror of point, point, point is pit, pit, pit and vice versa. If the two signatures are equivalent up to rotation, their mirror images are as well, and so the concept of mirror images extends to shapes. The main result of the theory of the patience game is as follows:

定理。对于每个块 B，存在一个唯一的孔 H，使得 H 的形状是 B 的形状的镜像，并且 H 是唯一可以容纳 B 的孔。

Theorem. For each block B there exists a unique hole H such that the shape of H is the mirror image of the shape of B, and H is the only hole B can fit into.

这可以转化为一种简单的方法来确定哪个块适合哪个孔：

This can be turned into a simple method to determine which block fits in which hole:

1. 用手指绕着一个块来确定它的形状。

1. Run your finger around a block to determine its shape.

2. 用手指沿着每个孔移动，直到辨认出相应的镜子形状。

2. Run your finger around each hole until you recognize the corresponding mirror shape.

3. 一旦找到镜子形状，您就知道找到了正确的孔，并且可以将块放入其中。

3. Once the mirror shape is found, you know you have the right hole, and you can fit the block into it.

真正的快乐

Real Pleasure

与所有数学定义一样，我们对形状的定义可能看起来任意且难以处理。

As with all mathematical definitions, our definition of shapes may seem arbitrary and unwieldy.

好消息是，我们成功地用触觉体验的词汇来谈论形状，而不需要任何视觉感知。换句话说，我们找到了一种用盲人也能理解的语言来描述星星形状的方法。这无疑是一项了不起的成就。

The good news is that we’ve managed to talk about shapes using the vocabulary of tactile experience, without any reference to visual perception. In other words, we’ve found a way to describe the shape of a star in a language a person without sight could understand. This undoubtedly is a tremendous feat.

坏消息是，我们的定义很丑陋。它完全无法捕捉视觉体验的丰富性和美感，以及形状为我们代表的一切：它们的明显性、它们的普遍性、它们的不可逃避的存在，以及让我们爱上它们的一切。对于所有这些，我们的定义是一个非常糟糕的替代品。

The bad news is that our definition is ugly. It absolutely fails to capture the richness and beauty of the visual experience, and all that shapes represent for us: their obviousness, their universality, their unescapable presence, and everything that makes us love them. For all this our definition is a very poor substitute.

但认为我们已经完成了就太愚蠢了。我们才刚刚开始。要解开一个形状到底是什么的毛线球，我们的定义只是无数其他可能的开端中的一个糟糕的开端。没有什么能阻止我们更加努力地工作，创造一种语言，这种语言将越来越巧妙地捕捉到星星或多或少尖、或多或少拉长、或多或少扭曲等等的含义。

But it would be stupid to think we’ve finished. We’ve only just begun. To unravel the ball of yarn of what a shape really is, our definition is a poor beginning among innumerable other possible beginnings. Nothing prevents us from working harder and crafting a language that will capture with more and more finesse what it means for a star to be more or less pointed, more or less elongated, more or less distorted, and so on.

然而，扩大词汇量、增加精确度和细节并不能解决问题。我们的问题更加深刻。观察不是一个词语的问题。视觉是一种感官的、本能的体验，我们无需思考就可以生活。

Yet extending the vocabulary, adding precision and detail, won’t fix the issue. Our problem runs deeper. Seeing isn’t a question of words. Seeing is a sensory, instinctive experience that we live without having to think about.

说一个形状是旋转的等价类特征，说一个星形是通过重复n次“点、坑”模式获得特征的形状，这很聪明，但永远不能完全令人满意。我们不是机器人。我们不希望通过一种似乎是反乌托邦官僚发明的语言来理解世界。我们想“看”而不去思考。

Saying that a shape is an equivalence class of signatures up to rotation, saying that a star is a shape whose signature is obtained by repeating n times the pattern “point, pit,” is clever enough, but it can never be entirely satisfactory. We’re not robots. We have no wish to apprehend the world through a language that seems to have been invented by dystopian bureaucrats. We want to “see” without thinking about it.

当你对用数学官方语言写成的文本感到困惑时，你就处于一个盲人的情况，他在无法看到星星的情况下制定我们对星星的正式定义。只要你无法直观地理解定义并理解“字里行间的想法”，这绝对是胡言乱语，至少一开始是这样。

When you puzzle over a text written in the official language of mathematics, you’re in the situation of a sightless person who is working out our formal definition of stars without being able to see them. It’s absolute gibberish, at least at first, as long as you’re unable to get an intuitive sense of the definition and access “the thoughts between the lines.”

数学理解正是如此：找到在自己内心创造正确的心理图像的方法，以代替正式的定义，将这种定义变成直观的东西，以“感受”它真正所说的内容。

Mathematical comprehension is precisely this: finding the means of creating within yourself the right mental images in place of a formal definition, to turn this definition into something intuitive, to “feel” what it is really talking about.

理解一篇数学文本，其中将星形定义为通过重复n次“点、坑”模式获得特征的形状，就是要让你忘记正式的定义，并直接感受到星形是什么，只需简单提及星形这个词即可。

Understanding a mathematical text that defines a star as a shape whose signature is obtained by repeating n times the pattern “point, pit,” is to get to a point where you forget the formal definition and sense directly what a star is, on command, with the simple mention of the word star.

数学的真正乐趣在于有一天醒来，意识到你可以在脑中看到星星，这是以前从未见过的。

The real pleasure of mathematics is waking up one day and realizing that you can see stars in your head, which you’d never been able to do before.

数学家的秘密技巧旨在促进和加速这种直觉理解。数学家使用逻辑和语言作为学习观察的工具。

The secret techniques of mathematicians aim to facilitate and accelerate this intuitive understanding. Mathematicians use logic and language as an apparatus for learning to see.

我知道这听起来好得令人难以置信，而且可能感觉远远超出了你的能力范围。然而事实并非如此。你有能力从抽象定义开始，直观地感受它所指代的内容。你已经做到了这一点。

I know that seems too good to be true, and it probably feels way out of your league. That is, however, not the case. You have the ability to start with an abstract definition and intuitively sense what it designates. You’ve already done this.

除了通过复杂的抽象数学概念组合之外，没有人向你描述过你在脑海中通过十亿减一就能想象出的数字。

No one has ever described to you, other than through a complex assemblage of abstract mathematical concepts, the number that you can picture in your head by taking a billion and subtracting one.

它的书写形式是 999,999,999，给人的印象是这个数字实际存在于你面前的页面上，但这只是一个极其复杂的正式定义的简写符号。它将这个数字描述为一系列加法和乘法的结果，如果你试图想象它，你会感到头疼：

Its written form as 999,999,999, which gives you the impression that the number is physically present on the page in front of you, is just the shorthand notation for an incredibly complex formal definition. It characterizes this number as the result of a chain of additions and multiplications that, if you tried to imagine it, would give you a headache:

九加九乘以十加九乘以十乘以九乘以十乘以九乘以十乘以十加九乘以十乘以十乘以十加九乘以十乘以十乘以十加九乘以十乘以十乘以十乘以十加九乘以十乘以十乘以十乘以十加九乘以十乘以十乘以十乘以十加九乘以十乘以十乘以十乘以十乘以十加十乘以十乘以十加九乘以十乘以十乘以十乘以十乘以十加十乘以十乘以十加九乘以十乘以十乘以十乘以十乘以十加十乘以十乘以十加十乘以十乘以十加九乘以十乘以十乘以十乘以十乘以十加十乘以十乘以十加十乘以十加十乘以十乘以十加九乘以十乘以十乘以十乘以十乘以十加十乘以十加十乘以十加十乘以十加十乘以十加九乘以十乘以十乘以十乘以十加 ...十乘以十加九乘以十乘以十乘以十乘以十加十乘以十加十乘以十加十乘以十加十乘以十加十乘

Nine plus nine times ten plus nine times ten times ten plus nine times ten times ten times ten plus nine times ten times ten times ten times ten plus nine times ten times ten times ten times ten times ten plus nine times ten times ten times ten times ten times ten times ten plus nine times ten times ten times ten times ten times ten times ten times ten plus nine times ten times ten times ten times ten times ten times ten times ten times ten.

在纸面上，这个数字是一个抽象的组合，逻辑清晰，冷冰冰。然而在你的脑海里，它却是一个简单的物体，具体而清晰。

On paper, this number is an abstract assemblage, logical and cold. Yet in your head, it’s a simple object, concrete and clear as day.

9

这里发生了一些事情

9

Something’s Going on Here

在我上学期间，经常有人对我说我握笔不正确，所以我写字像猪一样。

During my school years, I was often told that I wasn’t holding my pen correctly, and that’s why I wrote like a pig.

我选择学习数学是因为我认为他们会教我如何正确地在脑子里“记住”数学。我按照自己的方式做，效果也不错，但我从来都不确定我的方式是否正确。

I chose to study math because I thought that they’d teach me how to “hold” math correctly in my head. I did it my own way and that worked well enough, but I was never at all certain that my way was the right one.

在我的学习和科学生涯中，我最惊讶的是从未接受过任何有关该学科的正规教育，好像它不严肃或不值得花时间。

My biggest surprise, throughout my studies and scientific career, was never having received any formal education in the subject, as if it weren’t serious or worth the time.

毫无疑问，我有点天真，但在我看来，数学的核心问题不是某某定理是否正确，而是为什么它对某些人来说如此简单，而对其他人来说却如此困难。高中毕业后，在我读本科的第一年，我以为第一堂课会专注于如何在脑海中操纵数学概念的正确方法。他们肯定会从解释如何做到这一点开始！

I was, no doubt, a bit naïve, but it seemed to me that the central problem in math wasn’t whether such and such theorem was true, but why it was so easy for some and so difficult for others. After high school, in my first year as an undergrad, I was expecting that the first class would focus on the right way to mentally manipulate mathematical concepts. Surely they’d begin by explaining how to do it!

但第一节课的主题不同。在正式数学中，起点不是你在脑海中进行的看不见的动作，而是形式逻辑和集合论。我等待的解释并没有在下一节课或再下一节课中出现。最后我放弃了等待。

But the first class was on a different subject. In official math, the starting point isn’t the unseen actions that you do in your head, it’s formal logic and set theory. The explanation I was waiting for didn’t come in the next class, or the one after that. In the end I gave up waiting for it.

然而，几周后，当我们开始研究任意维向量空间时，这个问题又出现了。从那时起，我就开始认真思考这个问题。

The question, however, came up again a few weeks later, as we began the study of vector spaces in arbitrary dimensions. That’s when I became seriously preoccupied with it.

一维向量空间只是一条线。二维向量空间是一个平面。我们生活在三维向量空间中，或者更确切地说，我们倾向于相信我们生活在三维向量空间中，即使爱因斯坦向我们展示了事实并非完全如此。

A one-dimensional vector space is just a line. A two-dimensional vector space is a plane. We live in a three-dimensional vector space, or rather we have the tendency to believe we do, even if Einstein showed us how that’s not entirely true.

没有理由止步于三维空间。有了逻辑形式主义，你就可以继续前进。你可以定义四维空间、五维空间、六维空间等等。如果你愿意，你可以在 24 维、196,883 维或n 维（其中n是任意整数）上进行几何学研究。

There’s no reason to stop at three. With logical formalism, you can keep going. You can define what a four-dimensional space is, five-dimensional, six-dimensional, and so on. If you wanted to, you could do geometry in 24 dimensions, or 196,883 dimensions, or dimension n, where n is any whole number.

这些空间并不是实验室里的奇观。它们是基本概念，没有它们我们就无法理解周围的世界，它们对现代科学和技术如此重要，以至于一个多世纪以来，它们已经成为基本词汇的一部分，就像整数一样。

These spaces aren’t laboratory curiosities. They are fundamental concepts, without which we can’t understand the world around us, and they are so central to modern science and technology that for more than a century they’ve become part of the basic vocabulary, like whole numbers.

如果你从未学会多维度思考，你就错过了人生最大的乐趣之一。就像你从未见过大海，或从未吃过巧克力一样。

If you’ve never learned to think in multiple dimensions, you’ve missed out on one of the great joys of life. It’s like you’ve never seen the ocean, or never eaten chocolate.

想象力的成果

The Fruit of Your Imagination

当你在二维或三维空间中做几何学时，有一个简单的方法来展示你正在谈论的内容：画图。例如，在三维空间中，你可以把二十个等边三角形拼在一起，做成一个二十面骰子，就像图中那样。

When you do geometry in two or three dimensions, there’s an easy way to show what you’re talking about: make a drawing. For example, in a three-dimensional space, you can put together twenty equilateral triangles to make a twenty-sided die that looks like that pictured in the figure.

这种神奇的物体早已为人所知，被称为正二十面体。看着这幅画，你会觉得二十面体漂浮在空中。但这并不是你眼前的景象。你看到的是一张二维纸，上面画着二十面体的图像。更准确地说，这个图像就是所谓的投影：它是虚拟二十面体（三维）的阴影（二维）。

This remarkable object, which has been known for ages, is called a regular icosahedron. Looking at the drawing, you have the impression of seeing an icosahedron floating in space. But that’s not really what’s in front of your eyes. What you’re looking at is a two-dimensional page on which is found the image of an icosahedron. More precisely, this image is what is called a projection: it’s the shadow (in two dimensions) of an imaginary icosahedron (in three dimensions).

你的大脑可以很容易地从二维投影中重建三维物体。当你看你的假期照片时，你会感觉自己真的看到了三维场景。这不需要任何特殊的努力。它不会让你疲惫不堪，也不会引起任何形而上学的担忧。你永远不会相信这些场景实际上是在二维空间中发生的。你永远不会觉得你的三维感知不过是一种抽象，一种心理重建，仅仅是你想象的产物。你永远不会觉得你在照片中看到的是一种幻觉。

Your mind easily reconstructs three-dimensional things from their two-dimensional projections. When you look at your vacation photos, you have the sense of actually seeing scenes that occur in three dimensions. It doesn’t require any particular effort. It doesn’t tire you out or raise any metaphysical concerns. You never believe that the scenes are actually taking place in two dimensions. You never get the impression that your three-dimensional perception is nothing but an abstraction, a mental reconstruction that is simply the fruit of your imagination. You never have the sense that what you see in the photos is a hallucination.

你的大脑甚至可以看到图像中没有显示的东西。看着二十面体的投影，你不仅可以看到它，还可以在脑海中将其旋转，即使这需要一点注意力。实际的绘图保持完全静止。但这并不妨碍你轻松理解我所说的在脑海中“旋转”二十面体的意思。

Your mind can even see things the image doesn’t show. Looking at the projection of the icosahedron, not only can you see it, but you can turn it around in your head, even if that requires a bit of concentration. The actual drawing stays perfectly still. But that doesn’t stop you from easily understanding what I mean by “turning” the icosahedron in your head.

例如，如果你把二十面体旋转五分之一圈沿垂直轴旋转，你会发现和你开始时相同的二十面体。旋转不变性是二十面体的一个众所周知的性质。

For example, if you turn the icosahedron a fifth of the way around its vertical axis, you’ll find the same icosahedron you started with. Rotational invariance is a well-known property of icosahedrons.

如果我简单地将正二十面体定义为二十个等边三角形的抽象组合，而不给你画出它的方法，你会很难理解这种旋转不变性。有了图画，就容易多了。

If I had simply defined a regular icosahedron as an abstract assemblage of twenty equilateral triangles, without having given you the means to picture it, you would have had a lot more difficulty understanding this rotational invariance. With the drawing, it’s a lot easier.

视觉直觉使某些数学属性变得清晰，如果没有心理意象，这些属性根本就不清楚。这就是为什么将数学定义转化为心理意象如此重要。当你无法想象数学对象时，你会觉得自己并没有真正理解它们。你是对的。

Visual intuition makes certain mathematical properties clear, that without the mental image wouldn’t be clear at all. This is why transforming mathematical definitions into mental images is so important. When you’re unable to imagine mathematical objects, you have the sense that you don’t really understand them. And you’d be right.

盲人的几何学

Geometry for the Sightless

当你第一次听到有人谈论四维几何时，你会问自己这个第四维是什么。是时间吗？还是其他什么东西？

When you hear someone talk about four-dimensional geometry for the first time, you ask yourself what this fourth dimension could be. Is it time? Something else?

正确答案是，第四维度就是任何你想要的维度。

The right answer is that the fourth dimension is whatever you want it to be.

当你在平面上的二维空间中进行几何计算时，一个点由两个坐标决定，通常称为x和y，它们精确地表示了你想要它们表示的内容。

When you do geometry in two dimensions, on a plane, a point is determined by two coordinates, generally called x and y, that represent exactly what you want them to represent.

—当您看地图时，x通常是经度，y是纬度。

—When you look at a map, x is generally the longitude and y is the latitude.

— 当你绘制建筑物的正面时，x通常表示长度，y表示高度。

—When you draw the façade of a building, x is generally the length and y the height.

—当你表示兔子种群的增长时，x通常是时间，y是兔子的数量。

—When you represent the growth of a population of rabbits, x is generally time and y the number of rabbits.

同样的，在十维空间中，一个点由十个坐标决定，这些坐标一般被称为 x 1 , . . . , x 10。如果你想让这些坐标表示某种东西，它可以是任何你想要的。

In the same way, in a space with ten dimensions, a point is determined by ten coordinates that are generally called x1, . . . , x10. If you want these coordinates to represent something, it can be whatever you want.

如果你想描述入侵兔子种群的地理扩张，你需要从四个维度来思考，因为你需要四个坐标：经度、纬度、时间和种群密度。

If you wanted to describe the geographic expansion of an invasive population of rabbits, you would need to think in four dimensions, since you’d need four coordinates: longitude, latitude, time, and population density.

说四维几何是一种抽象概念是正确的。但它是一种简单而自然的概念。你的大脑可以接受第四维，甚至认为它是具体的，就像它接受所有那些实际上根本不具体的东西是具体的一样。入侵兔子种群的地理扩张是一个抽象的概念。如果它看起来很具体，那是因为你的大脑已经接受了第四维存在并且它是具体的这一观点。

It’s correct to say that four-dimensional geometry is an abstraction. But it’s a simple and natural one. Your mind can accept the fourth dimension and even find it concrete, the same way that it accepts as concrete all those things that in reality aren’t at all concrete. The geographic expansion of an invasive population of rabbits is an abstract notion. If it seems concrete, it’s because your mind has already accepted the idea that the fourth dimension exists and that it’s concrete.

与普遍看法相反，数学难懂的原因绝不是抽象。抽象是我们的普遍思维模式。我们使用的词语都是抽象的。说话、造句，就是操纵和组合抽象。从这个意义上讲，四维几何并不比二维几何更抽象。几何。四维几何的问题与抽象无关。问题在于它很难形象化，也很难画出来。

Contrary to common belief, it’s never abstraction that makes math difficult to understand. Abstraction is our universal mode of thinking. The words that we use are all abstractions. Speaking, making sentences, is to manipulate and assemble abstractions. In that respect, four-dimensional geometry isn’t any more abstract than two-dimensional geometry. The problem with four-dimensional geometry has nothing to do with abstraction. The problem is that it’s hard to visualize and hard to draw.

高维几何课程是为盲人开设的几何课程。它们类似于上一章讨论的触觉理论：它们不依赖视觉线索，而是使用数学语言和形式主义来定义几何词汇，其含义非常精确，但视觉解释并不简单。一切都可以用基于坐标的公式来描述。例如，有一个公式可以根据两点的坐标定义两点之间的距离。

Geometry courses on higher dimensions are geometry courses for the sightless. They resemble the theory of touch discussed in the previous chapter: instead of relying on visual cues, they use mathematical language and formalism to define a geometric vocabulary whose meaning is very precise but whose visual interpretation isn’t straightforward. Everything can be described using formulas based on coordinates. For example, there is a formula that defines the distance between two points from their coordinates.

起初，我们的大脑并不习惯这些新词汇。我们无法赋予它直观的视觉意义。这就是为什么四维几何不能像二维几何那样教授，因为在二维几何中，图形和直接的视觉直觉起着核心作用。

At first, our minds aren’t used to this new vocabulary. We can’t give it an intuitive visual meaning. This is why four-dimensional geometry can’t be taught in the same way as two-dimensional geometry, where figures and direct visual intuition play a central role.

例如，有一个与二十面体类似的四维物体。它是一个有 600 条边的骰子，非常规则，甚至比二十面体更漂亮。或者更确切地说，你应该说这个物体是一个有 600 条“超边”的“超骰子”。这些超边是三维物体，是正四面体，也就是有三角形底座的正金字塔。每个超边有 4 条边，这些边是等边三角形，它沿着这些边连接到其他 4 条超边。总共有 600 条超边、1,200 条边、720 条边（三角形的边）和 120 个顶点。

For example, there’s a four-dimensional analogue to the icosahedron. It is a die with 600 sides, insanely regular and even more beautiful than the icosahedron. Or rather, you should say that this object is a “hyper-die” with 600 “hyper-sides.” These hyper-sides are three-dimensional objects that are regular tetrahedrons, that is, regular pyramids with a triangular base. Each hyper-side has 4 sides that are equilateral triangles, along which it is attached to 4 other hyper-sides. In total there are 600 hyper-sides, 1,200 sides, 720 edges (the sides of the triangles) and 120 vertices.

您在想象中遇到了问题吗？如果有帮助的话，我会提供一张图纸。

Are you having problems visualizing it? If it’s any help, I’m providing a drawing.

该图片是四维超骰子的二维阴影（或者更确切地说是它的一个阴影，因为物体的阴影会根据其朝向光的方向而变化）。

The picture is the two-dimensional shadow of the four-dimensional hyper-die (or rather one of its shadows, since an object’s shadow changes according to its orientation toward the direction of the light).

看着这幅图会让人感觉很舒服，而且不会产生任何影响堡垒，看看你面前漂浮的“超二十面体”，悬浮在四维空间中。

It would be agreeable to look at this image and, without any effort, see the “hyper-icosahedron” floating in front of you, suspended in four-dimensional space.

我也想看到它在我眼前飘浮。我想一眼就抓住它，感受它的多维厚度。可惜，事实并非如此。

I’d also like to see it floating before my eyes. I’d like to grasp it at a glance and perceive its multidimensional thickness. Sadly, that’s not what happens.

我的大脑无法立即轻松地根据二维阴影在脑海中构建出四维物体的图像。我确实能够感知到超二十面体的物理存在，但我采用的方法不同，不依赖于绘图。

My mind isn’t capable of instantly and effortlessly constructing a mental image of a four-dimensional object from its two-dimensional shadow. I do manage to perceive the physical presence of the hyper-icosahedron, but I do it in a different way, without relying on the drawing.

真正的虚假图像

Truly False Images

在学习数学的过程中，我很快意识到我和其他人一样，我不知道如何像在二维或三维空间中那样看待高维几何物体。

In studying math, I quickly realized that I was just like everyone else, and I didn’t know how to see higher-dimensional geometric objects the same way I could in two or three dimensions.

但我也注意到了另一个相当微妙和出乎意料的现象。

But I also noticed another phenomenon, quite subtle and unexpected.

这种现象并不是什么特别的。它发生在幕后，在背景中，很容易被忽视。也许它一直存在而我却没有注意到。

This phenomenon wasn’t anything special. It was going on behind the scenes, in the background, and could easily have gone unnoticed. Maybe it had been going on forever without my even noticing.

无论如何，直到此时此刻，在我十八岁生日的前几周，当我们开始研究高维几何时，我才意识到这一点：我所学到的抽象几何概念在我心中唤起了或多或少视觉化的奇异印象。

At any rate, it was only at this very moment, a few weeks before my eighteenth birthday, as we had begun looking at higher-dimensional geometry, that I became aware of it: the abstract geometric notions that I had been taught evoked in me bizarre impressions that were more or less visual.

这些印象非常微弱，含义不明。它们是转瞬即逝的心理意象：混乱、模糊、转瞬即逝。它们也不稳定。有时它们存在，有时不存在。当它们存在时，这些意象总是幼稚的。更糟糕的是，它们总是虚假的。

These impressions were very faint and their meaning was unclear. They were fleeting mental images: confused, vague, and evanescent. They were also unstable. Sometimes they were there, sometimes not. When they were there, the images were always naïve. And what’s worse, they were always false.

这就像我的大脑试图通过拼凑二维和三维的心理图像来观察高维几何。结果却相差甚远。这些图像不只是有点假，就像画出的圆是假的，因为它不是完美的圆形。我的心理图像极其虚假。

It was as if my mind were trying to see higher-dimensional geometry by cobbling together two- and three-dimensional mental images. The result was ridiculously far off the mark. The images weren’t just a little false, like a circle that’s drawn is false because it’s not perfectly round. My mental images were grotesquely false.

我在巴黎最负盛名的预科学校路易大帝上学。我学习的是正统数学，包括公理、定义、命题、定理、证明、符号和公式。教学逻辑严谨，条理清晰。我学会了严谨而精确地写数学题。

I was studying at Louis-le-Grand, the most prestigious preparatory school in Paris. I was being taught serious math, the official mathematics with all its axioms, definitions, propositions, theorems, proofs, symbols, and formulas. The teaching was logical and structured. I was taught to write math rigorously and precisely.

然而，不知为何，我那天真的直觉并没有完全放弃。它似乎拒绝变得无关紧要。它根本不起作用，并给出了奇怪的结果。

And yet, for some reason, my naïve intuition hadn’t entirely given up. It was as if it was refusing to sink into irrelevance. It didn’t work at all and gave bizarre results.

我脑子里浮现的图案就像学龄前儿童的涂鸦，比如当我画出人们的手臂和腿直接连在头上时，却没有意识到我忘记了身体的一个重要部位。或者更确切地说，以一种困惑和混乱的方式意识到我忘记了一件重要的事情，却没有弄清楚它是什么可能是。我知道有些事情不对劲，但我说不出到底是什么。

The designs in my head resembled preschool scribblings, like when I drew people with their arms and legs attached directly to the head, without realizing I was forgetting a significant part of the body. Or rather realizing, in a confused and mixed-up manner, that I was forgetting something of importance, without figuring out what it might have been. Something was wrong, I knew, it but I couldn’t say what it was.

我清楚地记得四五岁时的情景。有一天，我打电话给老师，告诉她我的画有问题。这里面肯定有事，但到底是怎么回事？她回答说一切都很好，我的画其实很好。我清楚地感觉到她在取笑我。

I have this clear memory from when I was four or five. One day, I called on the teacher to tell her there was an issue with my drawing. Something going’s on here, but what? She replied that everything was fine, that my drawing was actually very nice. I had the very distinct impression she was making fun of me.

我并不想重演这段经历，也不想举手说我有问题，因为我脑子里的形象是假的。我也不想在众人面前被嘲笑。我的本能反应是把这些虚假的心理形象当成必须摆脱的寄生想法。

I had no desire to replay this experience and raise my hand to say that I had a problem because the images in my head were false. I had no desire to be ridiculed in front of everyone. My reflex was to treat these false mental images like parasitical thoughts that I had to get rid of.

如果路易大帝预科班成功的秘诀是像四岁小孩一样思考并在脑子里涂鸦，那么每个人都会明白。

If the secret of success in the preparatory class at Louis-le-Grand was to think like a four-year-old and make scribbles in your head, everyone would be onto it.

坚持用简单和形象的眼光看待事物是没有意义的。是时候长大了，学会如何用复杂而严肃的语言有条理地、有逻辑地思考。我需要像个成年人一样行事。

It made no sense to insist on seeing things simply and in images. It was time to grow up and learn how to think logically, in an organized manner, using complex and serious words. I needed to act like an adult.

更大或更小的管道

Bigger or Smaller Pipes

当时，我仍然相信逻辑可以帮助你思考。我自己无法逻辑思考，但我认为这是因为我出了问题。我相信学习数学可以帮助我解决这个问题，而第一步就是摆脱我天真而虚假的心理形象。

At the time, I still believed that logic helped you to think. I wasn’t myself able to think logically, but I thought it was because something was wrong with me. I believed that studying math would help me fix this issue, and that the first step was to get rid of my naïve and false mental images.

但是在所有这些虚假的图像中，在所有这些我想要摆脱的寄生思想中，我惊讶地发现了一个比其他的更不虚假的思想。

But amid all these false images, amid these parasitical thoughts that I wanted to get rid of, I was surprised to find one that was less false than the others.

当你研究向量空间时，你也会研究线性映射、核、秩、维数和余维数。向量空间通常用字母表示，线性映射用连接这些字母的箭头表示。但是当我选择将向量空间描绘成更大或更小的桶（根据其维度），将线性映射描绘成连接这些桶的更大或更小的管道（根据其秩）时，这些概念的所有练习突然变得显而易见。

When you study vector spaces, you also study the concepts of linear maps, kernel, rank, dimension, and codimension. Vector spaces are usually represented by letters and linear maps by arrows connecting these letters. But when I chose to picture vector spaces as bigger or smaller barrels (according to their dimension) and linear maps as bigger or smaller pipes (according to their rank) connecting those barrels, then all the exercises on these concepts suddenly became obvious.

没什么。那是一门很小的学科，还有很多我无法解决的练习。但在这门学科中，我不仅能够解决练习，而且它们对我来说变得像 1,000,000,000 – 1 = 999,999,999 一样明显。它们变得如此明显，以至于你甚至不得不问它们似乎很荒谬，更荒谬的是有人不知道如何解决它们。

It wasn’t much. That was a tiny subject and there were plenty of other exercises that I was unable to solve. But in this subject, not only was I able to solve the exercises, they became as obvious to me as 1,000,000,000 – 1 = 999,999,999. They became so obvious that it seemed absurd that you’d even have to ask, and even more absurd that there were people who didn’t know how to solve them.

这些桶和管子让我的生活变得简单，但它们是从哪里来的呢？你应该这样做吗？其他人的脑子里在想什么？他们是如何想象数学概念的？

These barrels and pipes made my life simpler, but where did they come from? Was that how you’re supposed to do it? What was going on in other people’s heads? How were they imagining mathematical concepts?

我记得我困惑地看着我的同学，仔细观察他们的脸，想看看他们脑子里在想什么。

I remember looking perplexedly at my classmates, scrutinizing their faces to look for signs of what was going on inside their heads.

我震惊地发现我根本就不知道。

I was shocked to find out I hadn’t the slightest idea.

一个巨大的问题

A Gigantic Problem

没有人告诉我们应该在脑子里做什么，而这正在成为一个巨大的问题。

No one had told us what we were supposed to be doing inside our heads, and that was becoming a gigantic problem.

我意识到，我们对所接受的指令有两种截然不同的看法，而且这两种方法互相不兼容。

I realized that there were two radically different ways of looking at the instruction we were receiving, and that the two approaches were mutually incompatible.

第一种方法是将数学视为知识。数学陈述是你必须吸收并能够记住的信息。你必须学习定义、学习定理、学习证明。

The first approach consists of treating mathematics as knowledge. Mathematical statements are information that you have to ingest and be able to recall. You have to learn the definitions, learn the theorems, learn the proofs.

第二种方法是拒绝学习。它把数学当成一种感官体验。数学陈述的唯一功能是帮助你产生心理意象，只有这些意象才能让你理解。一旦你有了正确的心理意象，其他一切都会变得清晰起来。

The second approach consists of refusing to learn. It takes on math as a sensory experience. The sole function of mathematical statements is to help you generate mental images, and only these images will lead to comprehension. Once you have the correct mental images, everything else becomes clear.

这两种方法是不相容的，因为它们需要相反的心态。用心学习，接受相信你不理解的东西——这只存在于第一种方法中。在第二种方法中，你会用怀疑和怀疑的眼光看待你不理解的东西：“真的吗？这应该是真的吗？不可能！这怎么可能？我怎么能想象得到呢？”

The two approaches are incompatible because they require opposite mindsets. Learning by heart, accepting to believe what you don’t understand—that only exists in the first approach. In the second approach, you look at what you don’t understand with distrust and incredulity: “Really? This is supposed to be true? No way! How’s that possible? How can I picture that?”

在那之前，我一直本能地遵循第二种方法，而且这种方法对我来说相当有效。在小学时，当老师解释什么是圆时，我事先就知道如何在脑海中描绘圆。这就是为什么数学对我来说如此简单。学校教会了我如何用语言来表达那些我已经很容易或多或少清晰地描绘的东西。

Until then, I had instinctively followed the second approach, and it had worked rather well for me. In primary school, when the teacher explained what a circle was, I knew beforehand how to picture circles in my head. It’s why math had been so easy for me. School had taught me how to put words to things that were already easy for me to picture more or less clearly.

但我已走到了那条路的尽头。现在我接触到了严肃而深刻的东西，我无法理解，我的直觉也无法想象。我已到达了心智能力的极限。这些更大或更小的管道的概念也许是我最后有效的直觉。甚至这也是运气问题。坦率地说，有了这些天真的形象，我还能期待什么呢？

But I was reaching the end of that road. I was now being introduced to serious and profound things that I couldn’t understand and that my intuition was incapable of visualizing. I was reaching the limits of my mental capacity. The concept of these bigger or smaller pipes was perhaps my final valid intuition. And even that was a matter of luck. With such naïve images, frankly, what more could I expect?

我无法依靠直觉，因此陷入了困境。我别无选择。终于到了我必须“学习”的时刻了。

And without being able to rely on my intuition, I was stuck. I didn’t have any choice. The moment had come when I finally had to “learn.”

但我很清楚其中的含义。把数学当成知识意味着放弃理解数学的乐趣，放弃感受数学在我心中活跃起来的乐趣。也意味着放弃对数学的热爱。

But I was well aware of the implications. Treating math as knowledge meant giving up the pleasure of understanding it, of feeling it come alive within me. It meant giving up loving it.

聆听不和谐之音

An Ear for Dissonance

说实话，这并不是数学第一次给我带来真正的困难。

To be honest, it wasn’t the first time math posed real difficulties for me.

初中时我们就已经开始用字母来表示数字了。“设n为整数。”但是如果n是整数，为什么不直接说出数字呢？为什么要这么神秘？他们就不能直接说出来吗？我有一种被排除在游戏之外的感觉，什么都不懂，不够聪明。

It had already happened at the beginning of middle school, when we had to use letters to designate numbers. “Let n be a whole number.” But if n was a whole number, why not just say the number? Why all this mystery? Couldn’t they just spit it out? I had the feeling of being left out of the game, of not understanding anything, not being smart enough.

当时，幸好这种情况没有持续太久。几周后，我不知不觉地接受了这样的想法：你可以用字母推理，也就是说，在不知道实际数字的情况下用数字推理。我明白，不知道实际数字才是关键。用字母推理是一种同时推理所有数字的方式。它是用有限数量的单词进行无限数量的计算。

At the time, thankfully, this didn’t last long. After a few weeks, without any conscious effort, I ended up accepting the idea that you could reason with letters, that is, reason with numbers without knowing the actual number. I understood that not knowing the actual number was in fact the whole point. Reasoning with letters was a way of reasoning with all numbers at once. It was doing an infinite number of computations with a finite number of words.

然而，在路易大学校，事情进展得更快，情况似乎难以克服。每周有十个新概念需要学习，而我完全不知道如何在脑子里安排它们。

At Louis-le-Grand, however, things were going much faster, and the situation looked unsurmountable. Each week there were ten new concepts to learn and I had no idea how to arrange them in my head.

就在我十八岁生日即将来临的这一刻，我做出了我科学生涯中最重要的决定，甚至可能是我一生中最重要的决定：我决定接受这些愚蠢的想法和寄生思想，而不是无视它们。我选择倾听它们，认真对待它们。

It was at this very moment, right before I turned eighteen, that I made the most significant decision in my scientific career, and possibly of my entire life: instead of ignoring my stupid ideas and parasitical thoughts, I decided to embrace them. I chose to listen to them and take them seriously.

当然，这并不意味着要信以为真。我很清楚它们是假的。这是显而易见的。但既然如此明显，我能确切地说出它们是假的吗？

Of course, that didn’t mean taking them at face value. I knew perfectly well that they were false. It was obvious. But since it was so obvious, was I able to say exactly in which way they were false?

今天，当我试图描述这种智力方法时，我是这样总结的：我开始倾听我的直觉与我的直觉之间的不和谐。和逻辑。在第 11 章中，我将用一个你应该能理解的例子来解释这在实际中意味着什么。

Today, when I try to describe this intellectual method, I sum it up like this: I began to listen to the dissonance between my intuition and logic. In chapter 11, I’ll explain what that means in practical terms with an example that should speak to you.

现在回想起来，我很惊讶自己竟然自己做出了这个决定，自己把所有事情拼凑起来，而没有人告诉我这是正确的做事方式。

In hindsight, I’m amazed that I made this decision on my own, piecing things together myself without anyone to tell me it was the right way of doing things.

我记得我曾尝试与同班同学泽维尔谈论这个问题。我的担忧与官方数学和我们所学的内容相差甚远，以至于我无法清楚地表达自己。我没有合适的词语来谈论这些问题。我花了几十年的时间才找到一种合适的方式谈论它们。

I remember trying to speak about it with a friend, Xavier, who was in the same class. My concerns were so far off from official math and what we were being taught that I failed to express myself clearly. I didn’t have the right words to talk about these issues. It took me decades to find a proper way to talk about them.

我没有理由相信这种方法会奏效，甚至也不指望它会奏效。这只是一个幼稚的实验。我纯粹是出于好奇心才这么做的，只是想看看，想知道它到底什么时候会失效，如何失效。我预计它会失败，因为似乎无法想象有一种理解数学的方法，而我们一直被蒙在鼓里。

I had no reason to believe that this method would work and I wasn’t even hoping that it would. It was just a childish experiment. I was carrying it out through sheer curiosity, just to see, to find out exactly when and how it would fall apart. I was expecting it to fail, because it seemed unthinkable that there was a method for understanding math and we had been kept in the dark.

然而，我很快发现我的方法奏效了。我越是思考这些愚蠢的想法，它们就越不愚蠢。我越是专注于寄生想法，它们就越清晰。我越是倾听直觉和逻辑之间的不一致，我就越能用语言来表达它。我的直觉从来都不是完美的，但它一直在进步，而我却不需要付出任何努力。

However, I soon found out that my approach worked. The more I thought about my stupid ideas, the less stupid they became. The more I focused on my parasitical thoughts, the clearer they became. The more I listened to the dissonance between my intuition and logic, the more I was able to transcribe it in words. My intuition was never perfect, but it kept progressing without any effort on my part.

几周后，我的学习方式就发生了变化。我开始以课堂作为我直觉的基准。我试图预测老师要说什么。大多数时候我都猜错了，但这让我明白我的直觉在哪些地方是正确的。我理解的东西，我理解得非常透彻，所以我可以依靠它们，而把注意力集中在其他东西上。

In the space of a few weeks, my way of studying was transformed. I began to use class as a benchmark for my intuition. I tried to predict what the teacher was going to say. Most of the time I got it wrong, but that let me figure out where my intuition was already correct. The things I understood, I understood so well that I could rely on them and concentrate on the others.

我不断地回想那些我不明白的事情，直到我明白了为什么我不明白。最终，我终于明白了。

I kept going back to what I didn’t understand until I understood why I didn’t understand it. And in the end that’s what allowed me to understand.

我们的普通直觉

Our Ordinary Intuition

只要学校拒绝教授数学的人类现实性，所有数学家都将自学成才。

As long as schools refuse to teach the human reality of mathematics, all mathematicians will be self-taught.

仅仅说数学是直觉的问题还不够。你还必须解释这种直觉是可以实现的，并描述你可以发展它的方法。没有什么比“数学直觉是一种只有少数人才能拥有的特殊能力”这个神话更令人生畏的了。

Saying that math is a matter of intuition isn’t enough. You also have to explain that this intuition is accessible and describe the ways that you can develop it. Nothing is more intimidating than the myth that mathematical intuition is something special that only a chosen few are endowed with.

数学直觉就是我们每天使用的直觉，但它是在与语言和逻辑的对抗中发展和巩固的。一旦我们不再相信它是上天的恩赐，而是努力改进它，我们的直觉就会变成这样。

Mathematical intuition is the same intuition we use every day, but developed and solidified by its confrontation with language and logic. It’s what our intuition becomes once we stop believing it’s a gift from heaven, and instead work on improving it.

数学常常让人感觉像园艺。除草、栽种、修剪、浇水。起初似乎什么都没有生长，但有一天你意识到它已经生长了。很难相信你可以从我们对空间的普遍看法开始，并发展它，使思考任何维度都成为本能。然而，事实就是如此。

Math often feels like gardening. You weed, plant, prune, water. It seems at first that nothing is growing, yet one day you realize that it has. It’s hard to believe that you can start with our common perception of space and develop it so that it becomes instinctive to think in whatever dimension. However, that’s the case.

在我们对数学智能的错误信念、迷信和压抑背后，隐藏着我们对心理可塑性及其规律的无知。我们将在下一章中继续讨论这个问题。

Behind our false beliefs about mathematical intelligence, behind our superstitions and inhibitions, there lies our ignorance about mental plasticity and the laws that govern it. We’ll come back to that in the next chapter.

为什么数学教育总是忽略这一点，对我来说仍然是个谜。好像老师们觉得自己没有资格谈论它。很少有人敢用简单的语言描述他们的直觉，并承认他们心智意象的幼稚。

Why math education consistently misses that point remains a mystery to me. It’s as if teachers felt they’re not qualified to talk about it. Very few people dare to describe their intuition in simple language and admit to the great naïveté of their mental images.

然而，一些最伟大的数学家却以令人不安的平静态度完成了这一任务。皮埃尔·德利涅就是一个引人注目的例子。他是格罗滕迪克最成功的学生，他本人也是一位非凡的数学家（格罗滕迪克曾经评价德利涅“他比我强”）。德利涅于 1978 年获得菲尔兹奖（因解决著名的韦伊猜想）并于 2013 年获得阿贝尔奖。

Nevertheless, some of the greatest mathematicians have done so with disconcerting tranquility. Pierre Deligne is a striking example. The most successful of Grothendieck’s students and himself an extraordinary mathematician (Grothendieck used to say of Deligne that “he’s better than me”), Deligne won the Fields Medal in 1978 (for solving the famous Weil conjectures) and the Abel Prize in 2013.

在 2013 年阿贝尔奖采访中，德利涅被问及他的工作风格。那是他谈论他的直觉和对高维几何的看法的时候。以下是他所说的内容：

In his 2013 Abel Prize interview, Deligne was asked about his working style. That was the occasion to talk about his intuition and his perception of higher-dimensional geometry. Here, among other things, is what he said:

能够猜测什么是真、什么是假，这很重要……

It’s important to be able to guess what is true, what is false. . . .

然后我就记不住哪些陈述是被证实的了。我试着在脑子里想象出一系列的画面。不止一幅画面，都是假的，但方式不同，我知道它们在哪些方面是假的。……

Then I don’t remember statements which are proved. I try to have a collection of pictures in my mind. More than one picture, all false but in different ways, and I know in which way they are false. . . .

这些图画非常简单。我只是在脑海里画出一个类似平面上的圆圈和一条扫过圆圈的移动线，但后来我知道这是假的，它不是一维的，而是高维的……

The pictures are very simple. I draw just in my mind something like a circle in the plane and a moving line which sweeps it, but then I know that this is false, that it’s not one-dimensional but higher dimensional. . . .

它总是由非常简单的图片组合而成。

It’s always very simple pictures, put together.

数学直觉是如此平庸、简单和愚蠢，你需要很大的自信才能不把它扔进垃圾桶。当你不再是孩子时，你只有一个愿望，那就是让你的错误直觉变得沉默。当我认为我必须摆脱愚蠢的想法和寄生思想时，我差点就实现了这个愿望。

Mathematical intuition is so banal, simple, and stupid, that you need a lot of self-confidence not to throw it in the trash. When you’re no longer a child, you have only one wish, and that’s to reduce your false intuitions to silence. That’s what almost happened to me when I thought I had to get rid of my stupid ideas and parasitical thoughts.

那个羞涩的小声音告诉你你不懂，那是你的数学直觉。不要将它与那个大声喧哗的声音混淆，后者告诉你你不够聪明。那个小声音会指引你。那是你需要倾听的声音。那是你需要用一生去照顾和保护的声音。

The shy little voice that’s telling you that you don’t understand, that’s your mathematical intuition. Don’t confuse it with the loud noisy voice that’s telling you that you’re not smart enough. The little voice will guide you. That’s the one you need to lend an ear to. That’s the one you need to take care of, and protect throughout your entire life.

10

观察的艺术

10

The Art of Seeing

当我思考高维几何物体时，我有足够清晰的视觉感知。但我看到它们的方式与看到物理世界中的物体的方式不同。我真的只看到某些方面、某些部分，以及我感兴趣的细节。我并不能真正看到它们的全部。但我用我的整个身体以一种弥散的方式感受到它们的存在。

When I think about higher-dimensional geometric objects, I have a clear enough visual perception. But I don’t see them in the same way I see objects in the physical world. I really only see certain aspects, certain pieces, the details that interest me. I’m not really able to see them entirely. But I sense their presence in a diffuse way, with my entire body.

据说，像看三维物体一样看四维或五维物体是不可能的。

Being able to see in four or five dimensions in the same way we see in three dimensions is said to be impossible.

尽管如此，比尔·瑟斯顿还是做到了。这种惊人的能力极其罕见。它甚至在数学界也引起了人们的钦佩，这也为他的传奇做出了贡献。

Nevertheless, Bill Thurston could do it. This amazing ability is exceedingly rare. It elicits admiration, even among the mathematical community, where it contributed to his legend.

对此感到害怕是正常的。人们很容易从中看出伟大的数学家与众不同，他们的大脑在生物学上比我们优越。然而，当你了解瑟斯顿的个人经历时，你就会意识到这种关于非凡天赋的假设并不成立。他的故事并不是一个天生就拥有非凡能力以五维视角看待世界的外星人的故事。

It’s natural to find that intimidating. It’s tempting to see in it the proof that the great mathematicians are different, with a brain that is biologically superior to our own. However, when you get to know Thurston’s personal history, you realize that this hypothesis of an extraordinary gift doesn’t hold. His story isn’t that of an alien endowed at birth with the extraordinary ability to see the world in five dimensions.

事实上恰恰相反。他的故事是关于一个天生残疾的小男孩，在他生命的最初几年里，他无法用三维空间来观察世界。

In fact it’s rather the opposite. His story is that of a young boy born with a handicap that during the first years of his life kept him from seeing the world in three dimensions.

瑟斯顿出生时患有严重的先天性斜视。他的两只眼睛的视野从未交汇。当他看物体，他每次只能用一只眼睛看。两个图像无法融合在一起，这使他无法直接感知深度。

Thurston was born with a strong congenital strabismus, or squint. The visual fields of his two eyes never met. When he looked at an object, he could see it with only one eye at a time. The two images couldn’t come together, which deprived him of any direct perception of depth.

他很幸运有一位慷慨而慈爱的母亲，从他出生起，她就投入了大量的时间和精力来帮助他克服残疾。当比尔两岁时，她与他一起工作了几个小时，向他展示了充满色彩和图案的特殊书籍，作为长时间再教育练习的借口。

He was fortunate to have a generous and loving mother, who from the time he was born devoted a lot of time and energy to help him overcome his handicap. When Bill was two years old she worked with him for hours, showing him special books filled with colors and designs, pretexts for long exercises of reeducation.

瑟斯顿对几何的热爱可以追溯到这一时期，这种热爱贯穿了他的一生。对空间、材料、纹理和形式的热爱贯穿了他的整个作品，并反映在他手稿中的精美图画中。

Thurston’s almost carnal love for geometry, a love that would continue throughout his life, traces back to this period. This love of space, materials, textures, and forms underpins the entirety of his work and is reflected in the marvelous drawings that illustrate his manuscripts.

上小学后，瑟斯顿下定决心每天努力拓展自己的想象能力。在很小的时候，他不知不觉就成为了一名非凡的数学家。

When he began primary school, Thurston resolved to work each day to expand his capacity for visualization. Very early in his life, without knowing it, he became an extraordinary mathematician.

认为他比其他孩子花了更多时间学习如何看东西是错误的。我们睁着眼睛观察周围世界的每一秒都在增强我们的视觉能力。学习看东西是我们生命最初几年的主要活动之一（而且不仅仅是最初几年）。但对于绝大多数孩子来说，这是一种无意识的活动。它不断地、在幕后、几乎无意识地发生，而且不需要太多的注意力。

It would be a mistake to believe that he spent more time than other children learning how to see. Every second we have our eyes open, as we observe the world around us, we’re building on our capacity for visualization. Learning to see is one of the main activities of the first years of our lives (and not only the first years). But for the vast majority of children it’s an unconscious activity. It happens continually, in the background, almost unintentionally, and without much concentration.

瑟斯顿没有这种奢侈。他不能简单地听天由命。对他来说，观察世界并不是出于本能。学习观察是一个有意识的项目。你甚至可以说，这是他一生的工作。

Thurston didn’t have this luxury. He couldn’t simply let nature take its course. For him, there was nothing instinctive about seeing the world. Learning to see was a conscious project. You could even say it became the work of a lifetime.

福斯贝里无法像其他人那样跳跃，所以他必须发明自己的跳跃方式。瑟斯顿无法像其他人那样看东西，所以他必须发明自己的观察方式。福斯贝里的方法使他能够跳得更高。像瑟斯顿那样刻意努力看得更清楚，可以让你看得更远、更清楚，直达事物的核心。

Fosbury wasn’t able to jump like others so he had to invent his own way of jumping. Thurston wasn’t able to see like others so he had to invent his own way of seeing. Fosbury’s approach allowed him to jump higher. Deliberately working to see better, as Thurston did, allows you to see further, more distinctly, down to the heart of things.

当我们认为自己直接看到的是三维世界时，我们会无意识地拼凑视网膜捕捉到的二维图像。这种感知空间的方式并不完美。它不是客观感知，而是主观感知，相对于我们所观察的地方，并受到透视效应的扭曲。更糟糕的是，从我们局部的角度来看，世界的大部分对我们来说都是隐藏的。

When we believe we’re directly seeing the world in three dimensions, we’re unconsciously piecing together the two-dimensional images captured by our retinas. This way of perceiving space is imperfect. It’s not an objective perception but a subjective one, relative to the place we’re looking from and distorted by the effect of perspective. What’s worse, from our local perspective, the greater part of the world is hidden to us.

瑟斯顿无法轻易获得三维感知。他努力通过思维的力量，以自己的方式构建自己的三维感知。如果瑟斯顿有天赋，那就是耐心和决心。或者也许是爱和自信。

Thurston was denied this easy access to three-dimensional perception. He worked to construct his own, in his own way, through the power of thought. If Thurston had a gift, it was that of patience and determination. Or perhaps that of love and self-confidence.

数学工作并不是一系列灵光闪现的洞见和天才的灵感。它首先是一项基于重复同样的想象力练习的再教育工作。

Mathematical work isn’t a series of lightning insights and strokes of genius. It’s first of all a work of reeducation based on the repetition of the same exercises of imagination.

进步是缓慢的，因为身体需要时间来改变自己。强迫自己进步是没有用的，最终可能会伤害到你。你只需要坚持定期的训练计划，保持冷静，即使看起来你没有取得任何进步也要坚持下去。这就像去看语言治疗师或理疗师，只不过你一个人在想着自己的想法。

Progress is slow because the body needs time to transform itself. It doesn’t help to force it, which may end up hurting you. You just need to commit to a regular training schedule, keep your cool, keep going even when it seems you’re not making any progress. It’s like going to the speech or physical therapist, except you’re all alone and inside your head.

展望未来

Seeing Further

瑟斯顿有意识地、认真地发展他想象世界的能力。通过坚持不懈地在头脑中拼凑二维图像，他能够学会用三维来观察。

Thurston consciously and conscientiously developed his ability to picture the world. Through persistence, by working to stitch together two-dimensional images in his head, he was able to learn to see in three dimensions.

但为什么要止步于此呢？他意识到，利用同样的技术，他可以走得更远。通过把三维图像拼凑在一起，他学会了如何用四维来观察。通过把四维图像拼凑在一起，他学会了如何用五维来观察。

But why stop there? He realized that with the same technique he could go even further. By putting together images in three dimensions, he learned how to see in four. And by putting together images in four dimensions, he learned how to see in five.

即使只有三维空间，瑟斯顿的方法也能让他看到以前从未有人见过的东西。他于 1982 年提出的几何化猜想正是针对第三维空间的。猜想是一种数学陈述，有人认为它有效但尚未能够证明。做出猜想就是感觉某事是正确的，但无法说明原因。它本质上是一种富有远见和直觉的行为。

Even in only three dimensions, Thurston’s approach made him able to see things no one had seen before. His geometrization conjecture, formulated in 1982, deals precisely with the third dimension. A conjecture is a mathematical statement that someone believes is valid but isn’t yet able to prove. Making a conjecture is feeling something is right without being able to say why. It is by nature a visionary and intuitive act.

瑟斯顿猜想是一项重大突破。它尤其包含了著名的庞加莱猜想，该猜想于 1904 年提出，但长期未得到解决，因此在 2000 年制定的七大数学难题名单中被指定为“千禧年大奖难题”，每个难题的解答都会获得一百万美元的奖金。

Thurston’s conjecture was a spectacular breakthrough. It notably encompassed the famous Poincaré conjecture, which had been formulated in 1904 and remained unsolved for so long that it was designated a “Millennium Prize Problem” in a list established in 2000 of seven mathematical problems judged the most profound and difficult, each with a million-dollar award for its solution.

2003 年，格里沙·佩雷尔曼证明了瑟斯顿的猜想事实。因此，他也能够解决庞加莱猜想。我们将在第 17 章中进一步讨论佩雷尔曼和百万美元。

In 2003 Grisha Perelman was able to prove Thurston’s conjecture. He was thus also able to solve Poincaré’s conjecture. We’ll talk more about Perelman and the million dollars in chapter 17.

眼见为实

Seeing Is Believing

当瑟斯顿声称他能看到四维或五维空间时，他的意思是什么？他到底看到了什么？我们应该从什么意义上理解他使用的动词“看到”？

When Thurston claimed to see in four or five dimensions, what did he mean? What was he seeing exactly, and in what sense should we understand his use of the verb to see?

回答这些问题的最好方式就是把它们反过来想。我们到底看到了什么？我们所说的“看到”是什么意思？

The best way to answer these questions is to turn them around. What exactly do we see? What do we mean by “see”?

在使用动词“自己看”时，我们倾向于高估其含义。当我们环顾四周时，我们会产生一种与世界直接联系的幻觉，仿佛我们的眼睛是直接钻进我们意识的魔法窗户，让我们直接接触现实。如果我们想赋予动词“看”这样的含义，那么我们必须准备好面对后果：在这种情况下，我们实际上什么也没看到，我们只是相信我们看到了。

In using the verb to see for ourselves, we have the tendency to overestimate its meaning. When we look around us, we have the illusion of a direct relation to the world, as if our eyes were magic windows drilled directly into our consciousness and giving us direct access to reality. If that’s the meaning we want to give to the verb to see, then we have to be ready to face the consequences: in this case, we never really see anything, we just believe that we do.

你所看到的从来都不是未经修饰的现实，而是对世界的一种解读。换句话说，它是一种重建，由你的记忆和想象产生，基于你从未直接意识到的原始视觉信号。你出生那天，你还不知道如何看，因为你的大脑还没有学会理解视觉神经传递的原始信息。

What you see is never really unvarnished reality but an interpretation of the world. In other words, it’s a reconstruction, produced by your memory and imagination, based on raw visual signals that you’re never directly aware of. The day you were born, you didn’t yet know how to see, because your brain hadn’t yet learned to give a sense to the raw information delivered by your optic nerves.

这种重构能力现在让你能够想象不存在的事物，并产生看到它们的印象。看到某物和想象看到它并没有什么不同。你可以想象看到一只像马一样大的蚂蚁，就在你眼前。你可以想象它、描述它，甚至说出很多关于它的事情。但其他人没有办法直接看到你头脑中的巨型蚂蚁。

This same ability for reconstruction allows you now to imagine things that don’t exist, and to have the impression of seeing them. Seeing something and imagining seeing it aren’t that different. You can imagine seeing an ant as big as a horse, right there, before your eyes. You can picture it, describe it, and even say a lot of things about it. But others have no means of directly seeing the giant ant in your head.

颜色也是一样。红色给你一种非常特别的感觉。但你头脑中对红色的感觉到底是什么？你知道这和我的一样吗？也许这个问题本身就没有意义。

It’s the same with colors. Red gives you a very particular sensation. But what exactly is the sensation of red in your head? How do you know it’s the same as mine? Maybe that question itself makes no sense.

道尔顿和他的天竺葵

Dalton and His Geranium

我们在描述和传达我们的视觉感受方面非常困难，直到 1792 年秋天才有人意识到我们对于颜色的生物学感知并不平等：大约 8％ 的男性（和 0.6％ 的女性）是色盲。

We have such a hard time describing and communicating our visual sensations that it wasn’t until the autumn of 1792 that someone realized that we aren’t all equal in the biological perception of colors: around 8 percent of men (and 0.6 percent of women) are color-blind.

为什么如此惊人且易于证明的事实，自从我们的祖先开始谈论颜色以来，数千年来一直不为人知？

How is it that such a striking and easily shown fact could remain unknown for thousands of years, from time immemorial when our ancestors began to speak of color?

正是约翰·道尔顿，这位伟大的物理学家，将物质由原子组成的现代观念归功于他，他根据自己的案例做出了这一发现。他在一篇科学通讯中披露了他的发现，这篇通讯的标题耸人听闻，暴露了他个人的愚昧：“与色彩视觉有关的非凡事实。”

It was John Dalton, the great physicist to whom we owe the modern idea that matter is composed of atoms, who made the discovery based on his own case. He revealed his findings in a scientific communication whose sensationalist title betrays his own personal stupefaction: “Extraordinary Facts relating to the Vision of Colors.”

道尔顿似乎感到十分震惊，就好像他是第一个发现左撇子和右撇子的人一样。但当你读了他的叙述，你就会明白这种误解为何会持续数千年。

Dalton seems almost as astounded as if he had been the first to discover that there were right-handed and left-handed people. But when you read his account, you understand the mechanism that allowed this misunderstanding to persist for millennia.

从现代科学的角度来看，这个故事很简单。我们对颜色的感知是由视网膜中存在称为视锥细胞的专用细胞来解释的。人眼通常有三种视锥细胞，分别对蓝光、绿光和红光敏感。我们只能通过蓝色、绿色和红色的相对比例来感知各种颜色的细微差别。（正是由于这个简单的原因，屏幕在每个像素中结合了这三种颜色。）道尔顿是基因突变的携带者。他只有两种视锥细胞。他缺少对绿光敏感的视锥细胞，这使他无法感知某些细微差别。例如，他发现很难区分蓝色和粉色。但道尔顿成长在一个没有人想象过这种可能性的世界里，所以他或多或少学会了如何命名别人看到和谈论的所有颜色。他确信自己真的看到了它们。他看到了色彩斑斓的世界，从未怀疑过自己错过了什么。

From the point of view of modern science, the story is simple enough. Our perception of colors is explained by the presence within our retinas of dedicated cells called cones. A human eye normally has three types of cones, sensitive respectively to blue, green, and red light. We perceive a large palette of nuances of colors, but only through their relative proportion of blue, green, and red. (It’s for this simple reason that screens combine these three colors in each pixel.) Dalton was a carrier of a genetic mutation. He only had two types of cones. He was missing the cones sensitive to green light, which kept him from perceiving certain nuances. For example, he found it hard to distinguish blue from pink. But Dalton grew up in a world where no one imagined that was possible, so he learned more or less how to name all the colors that others saw and talked about. He convinced himself he was really seeing them. He saw the world in color, without ever suspecting that he was missing something.

然而，从道尔顿的角度来看，在能够用科学来解释的语言出现之前，他的故事读起来就像一出荒诞喜剧。

However, when told from Dalton’s point of view, before words were invented that allowed for its scientific explanation, his story reads like an absurdist comedy.

道尔顿在叙述的开头坦白道：他一生都觉得颜色的名字选得不太好。当人们有时用红色代替粉色时，道尔顿觉得这很荒谬。对他来说，粉色看起来更像蓝色，一点也不像红色。但他从来不敢告诉任何人。

Dalton begins his account with a confession: all his life, he’d had the impression that the names of colors weren’t well chosen. When people sometimes used the word red instead of pink, Dalton found it absurd. Pink, for him, looked more like blue, nothing at all like red. But he never dared to tell anyone.

1790 年，道尔顿开始对植物学产生兴趣。他很难辨别花的颜色，但这并没有让他太惊讶。他只是得到了帮助。当他问人们一朵花是蓝色还是粉色时，他从人们的眼神中看出他们认为这一定是某种恶作剧。他不明白为什么他们看起来很困惑，但从来没有花时间去弄清楚。他已经习惯了所有关于颜色的对话中的奇怪之处。如果道尔顿没有在 1792 年秋天发现一种具有绝对非凡特性的天竺葵，这种误解可能会永远持续下去。

In 1790, Dalton began to get interested in botany. He had a hard time recognizing the colors of flowers but that didn’t surprise him too much. He simply got help. When he asked people whether a flower was blue or pink, he saw in their eyes that they thought this had to be some kind of prank. He didn’t understand why they looked puzzled, but never took the time to find out. He was already accustomed to the strangeness in all conversations about color. The misunderstanding could have gone on forever if Dalton hadn’t made the discovery, in the autumn of 1792, of a geranium with absolutely extraordinary properties.

据说这株天竺葵是粉红色的。然而，在阳光下，它看起来是蓝色的。到目前为止，没有什么不寻常的：对道尔顿来说，这两种颜色总是非常接近。但当道尔顿想在烛光下看它时，天竺葵变成了鲜红色，对他来说，这种颜色与粉红色毫无关系。

This geranium was said to be pink. However, in sunlight, it looked blue. So far, nothing unusual: for Dalton, those two colors were always very close. But when Dalton had the idea to look at it by candlelight, the geranium turned bright red, a color that for him had nothing to do with pink.

道尔顿大吃一惊，于是带朋友过来欣赏他神奇的天竺葵。当他的朋友告诉他，他的天竺葵没有什么特别的，他很沮丧。似乎唯一能理解的人是他自己的兄弟。这就是道尔顿对颜色感知进行实验的起点，这些实验使他能够展示色盲（有时称为道尔顿病）及其遗传性。

Dalton was so flabbergasted that he brought his friends over to admire his miraculous geranium. When his friends told him that his geranium was nothing special, he was crestfallen. The only person who seemed to understand was his own brother. That was the starting point for Dalton’s experiments on the perception of colors that allowed him to demonstrate color blindness (sometimes called Daltonism) and its hereditary character.

这个“非凡”的故事有三个寓意。

There are three morals to this “extraordinary” story.

第一点是：在原始感知和我们认为我们看到的东西之间，有很大的回旋余地。道尔顿觉得颜色的名称很奇怪，但这并没有阻止他接受它们并以自己的方式解释它们。他构建了自己的颜色标度，与非色盲人的颜色标度不同，但并不一定更贫乏。最引人注目的例子是他对粉红色和红色之间细微差别的超敏反应。他比非色盲人更强烈地看到这些细微差别，就像一个对触觉和声音产生超敏反应的盲人一样（我们将在后面几页中讨论这一点）。

The first is this: between raw perception and what we believe we see, there’s a lot of room for maneuver. Dalton found the names of colors strange but that didn’t keep him from accepting them and interpreting them in his own way. He constructed his own color scale, different from non-color-blind people’s, but not necessarily more impoverished. The most striking example was his hyper-sensitivity to the nuances separating pink and red. He saw these nuances more intensely than a non-color-blind person, like a sightless person who develops a hyper-sensitivity to touch and sound (we’ll come back to this in a few pages).

第二个寓意与科学发现的过程有关。道尔顿的力量不在于强大的推理能力，而在于他能够感觉到某些事情不对劲，并且不停止寻找，直到他发现问题所在。在成为一项伟大的科学发现之前，这只是一种奇怪的感觉。数万年来，数十亿色盲患者都经历过同样的奇怪感觉，却无法用语言表达。

The second moral concerns the process of scientific discovery. Dalton’s strength wasn’t an enormous power of reasoning, but an ability to feel that something wasn’t right and not to stop searching until he’d found out what it was. Before it became a great scientific discovery, it was just a weird feeling. For tens of thousands of years, billions of color-blind people had experienced the same weird feeling without being able to put words to it.

这就是第三个道理。我们可以与那些与我们看法不同的人和平共处，而永远不会意识到差异。解释很简单：我们看不到他们的内心，实际上，我们看不到他们所看到的东西。

And that’s the third moral. We can peacefully coexist with people who don’t see the same things as us, without ever becoming aware of the differences. The explanation is simple: we don’t see inside their heads, and, literally, we don’t see what they see.

当道尔顿说蓝色类似于粉色时，道尔顿以外的人并不把他的话当真。当瑟斯顿说他能看到五维空间时，瑟斯顿以外的人也难以相信。

When Dalton said that blue resembled pink, non-Daltonians didn’t take him seriously. When Thurston said he saw in five dimensions, non-Thurstonians found it hard to believe.

有一个简单的方法可以确保色盲人士不会嘲笑你（或者，如果你是色盲，非色盲的人不会嘲笑你）：石原测试，使用由不同颜色的小圆圈组成的图像。你在图像中看到的东西不同，这取决于你是否认为某些圆圈是相同的颜色。在测试的其中一个盘子中，色盲者清楚地看到了数字 21，而其他人清楚地看到了 74。

There’s an easy way to make sure that color-blind people aren’t making fun of you (or, if you are color-blind, that non-color-blind people aren’t making fun of you): the Ishihara test, which uses images formed of small circles of different colors. You see something different in the images, depending on whether you perceive that certain circles are or aren’t the same color. In one of the plates in the test, color-blind people clearly see the number 21, whereas others clearly see 74.

没有针对想象力的石原测试。也没有直接的方法来验证一个人是否知道如何想象第五维度。

There’s no Ishihara test for the imagination. There’s no direct way to verify that someone knows how to visualize the fifth dimension.

我不知道瑟斯顿到底看到了什么，但当我看到他的数学著作时，我毫不怀疑他看到了很多我没有看到的东西。他的写作风格给人的印象是他只是想与我们分享它们。他很想在现实生活中直接展示这些事物，但他知道这是不可能的。所以他写数学论文。

I don’t know what Thurston really saw, but when I look at his mathematical work, I haven’t the slightest doubt that he saw many things I don’t. His style of writing gives the impression that he’s just trying to share them with us. He’d love to show the things directly, in real life, but he knows that’s impossible. So he writes math papers.

看见就是发现显而易见的东西

Seeing Is Finding Something Evident

在接受《纽约时报》采访时，瑟斯顿总结道：“人们不明白我如何能将四维或五维的事物形象化。五维形状很难形象化——但这并不意味着你不能思考它们。思考其实和看是一样的。”

In an interview with the New York Times, Thurston summed up things as follows: “People don’t understand how I can visualize in four or five dimensions. Five-dimensional shapes are hard to visualize—but it doesn’t mean you can’t think about them. Thinking is really the same as seeing.”

不过，最后一句话需要澄清一下。在下一章中，我们将讨论快速直观的思维方式与缓慢反思的思维方式之间的细微差别。对于数学家来说，“看见”意味着以快速直观的方式思考，直接思考，无需反思，就好像物体真的存在，就好像它就在你面前。

The last sentence, however, warrants some clarification. In the next chapter we’ll discuss the nuance between a rapid and intuitive way of thinking, and a slow and reflective way. For a mathematician, “seeing” signifies thinking in a rapid and intuitive manner, directly, without need for reflection, as if the object really existed, as if it were right there in front of you.

便捷性和即时性比感知的视觉性质更重要。看到就是发现显而易见的东西。从词源上来说，这是正确的（“明显”来自拉丁语videre，意思是“看到”）在日常生活中也是如此：当你看着一块冰时，虽然你不能直接看到它的温度，但显然它很冷。

Ease of access and immediacy count more than the visual nature of perception. Seeing is finding something evident. It’s true etymologically (“evident” comes from the latin videre, which means “to see”) and it’s also true in everyday life: when you look at a block of ice, it’s evident that it’s cold, although you can’t directly see its temperature.

聆听世界的耳朵

An Ear for the World

本·安德伍德 1992 年出生于加利福尼亚州。他两岁时，母亲发现他一只眼球后方有奇怪的反光。原来是视网膜癌。三岁时，他不得不接受双眼摘除手术。有时人们会用“视力受损”这个词来委婉地表达自己的意思。但在本·安德伍德的例子中，用委婉的说法来掩饰是没用的。他当时是盲人。

Ben Underwood was born in California in 1992. When he was only two years old his mother saw a strange reflection in the back of one of his eyes. It was retinal cancer. When he was three, he had to undergo an operation to remove both his eyes. Sometimes people use the term “visually impaired” as a polite euphemism. In Ben Underwood’s case, it’s useless to hide behind a euphemism. He was blind.

七岁时，本发现自己拥有一种魔力：通过咂舌，他就能看到周围的世界。

When he was seven, Ben discovered he had a magic power: by clicking his tongue, he was able to see the world around him.

真正的魔法并不存在。本·安德伍德只是学会了通过回声定位来观察，就像蝙蝠或海豚一样。咔嗒声是声纳信号。每个物体都会发回一个特征回声，告诉你它的位置、大小、形状以及它的成分。

Real magic doesn’t exist. Ben Underwood had simply learned to see by echolocation, like bats or dolphins. Clicks are sonar signals. Every object sends back a characteristic echo that tells you about its location, size, form, and what it’s made of.

我们已经知道回声可以告诉我们周围的空间。要分辨出浴室和大教堂内部的区别，你只需倾听。也许你已经有过这样的奇妙经历：某天早上醒来，你知道甚至还没睁开眼睛，你就已经发现夜里下雪了，因为寂静的质感已经改变了。

We already know that echoes can inform us about the space around us. To tell the difference between your bathroom and the inside of a cathedral, you just have to listen. And maybe you’ve already had the fascinating experience of waking up one morning and knowing, without even having opened your eyes, that it snowed during the night, because the texture of the silence has changed.

我们更难以相信的是，仅凭耳朵就能重建我们周围世界的可靠而详细的图像。然而，这正是本·安德伍德能够做到的。

What we have a much harder time believing is that it’s really possible, using only your ears, to reconstruct a reliable and detailed image of the world that surrounds us. It is, however, what Ben Underwood was able to do.

没有人向他解释如何做到这一点，甚至没有人告诉他这是可能的，但他却发展出了真正的“视觉”能力。视频显示他可以自由移动，无需使用拐杖或触摸物品。他可以做日常生活中的所有动作：上下楼梯、开门、不拿起面前的物体而说出它们的名字、在街上行走、指着树木和树枝、骑自行车、溜冰、绕着汽车转、打篮球。

Without anyone explaining how to do it, without anyone even telling him it was possible, he developed a genuine faculty of “vision.” Videos show him moving about freely, without using a cane or touching things. He does all the actions of daily life: going up and down stairs, opening doors, naming objects in front of him without picking them up, walking in the street, pointing at trees and their branches, riding a bike, roller-skating, weaving around cars, playing basketball.

本·安德伍德并不是第一个掌握回声定位的盲人。这一现象被人们所知和记录了近三百年。但在他之前，没有人能将其发挥到如此完美的程度。

Ben Underwood isn’t the first sightless person to develop echolocation. The phenomenon has been known and documented for nearly three hundred years. But no one before him had brought it to this high a degree of perfection.

本·安德伍德打破了人们认为人类可能达到的极限。他的能力使他在科学界和公众中闻名。十几岁时，他被邀请参加奥普拉·温弗瑞秀，分享他的故事。

Ben Underwood shattered the limits of what was believed humanly possible. His abilities made him famous among the scientific community as well as the general public. As a teenager, he was invited on The Oprah Winfrey Show to share his story.

如果他能够继续发展他的技术并继续分享他的秘密，他会取得什么成就？我们永远不会知道。本·安德伍德 16 岁时因癌症复发而去世。

What might he have accomplished had he been able to pursue the development of his technique and continued to share his secrets? We’ll never know. Ben Underwood died at age sixteen due to a recurrence of the cancer that had taken his sight.

心理可塑性的规律

The Laws of Mental Plasticity

比尔·瑟斯顿和本·安德伍德都是天才。但天才到底是什么？是智力的问题吗？好奇心？勇气？还是意志力？

Bill Thurston and Ben Underwood are striking geniuses. But what exactly is a genius? Is it a question of intelligence? Curiosity? Courage? Willpower?

我深深地钦佩瑟斯顿和安德伍德，但我讲述他们的故事不仅仅是为了分享我的钦佩。

I deeply admire both Thurston and Underwood, but it’s not only to share my admiration that I’m telling their stories.

真正的问题是我们对大脑功能的毫无根据的信念。我们幻想自己能够直接进入“真实”世界，而不受大脑构建的表征的影响。通过忽略我们拥有的大量可能性，我们为自己的智力设定了荒谬的限制。本·安德伍德的故事令人难以置信，我们本能地上网查了一下，确认这不是什么都市传说。这只是让我们略微了解一下我们拒绝接受的美妙体验。

The real subject is our unfounded beliefs about the functioning of our brain. It’s our illusion of being able to directly access the “real” world independently of the representation that our brain constructs of it. By ignoring the enormous range of possibilities available to us, we assign absurd limits to our intelligence. Ben Underwood’s story is so hard to believe that we have the reflex to check on the internet that it’s not some urban legend. That just gives a small idea of the fabulous experiences we deny ourselves.

我们的文化和教育明显没有教会我们关于人类非凡的心理可塑性，以及我们的命运在很大程度上取决于我们选择如何利用这种可塑性。数学教育的失败只是这一疏忽造成的附带损害。那些没有机会偶然重新发现利用这种可塑性为数学服务的行为的人注定永远无法理解它。

The glaring omission of our culture and of our education is to teach us about our extraordinary mental plasticity, and that our destiny depends in great part on what we choose to do with it. The failure of math education is simply collateral damage of this omission. Those who don’t have the chance to accidentally rediscover the actions that use this plasticity in the service of mathematics are condemned never to understand it.

我们普遍对心理可塑性的基本原理一无所知，这是远远超出数学范畴的巨大浪费。我并不假装知道或理解一切，但我认为以下是关键点：

Our general ignorance of the basic principles of mental plasticity is an enormous waste that reaches far beyond mathematics. Without pretending to know or understand everything, here are what seem to me the essential points:

1.我们心理可塑性的力量令人震惊，几乎是超自然的。

1. The power of our mental plasticity is profoundly shocking and almost supernatural.

比尔·瑟斯顿和本·安德伍德的故事总是令人难以置信。它们令人惊叹，但又带有一丝难以摆脱的耸人听闻的意味。你会问自己这是什么花招。但没有任何花招。从生物学的角度来看，这一切都很正常。

Stories like those of Bill Thurston and Ben Underwood are always hard to believe. They’re amazing, but have a hint of sensationalism that’s hard to shake off. You ask yourself what’s the gimmick. But there isn’t any gimmick. From a biological point of view, all of this is normal.

我们的怀疑有一个简单的解释，即工作机制的无意识性质。当有人告诉我们本·安德伍德通过分析他的大脑的回声来看待世界时我们可以想象他正在解普通人无法理解的复杂数学方程。这既是真的，也是假的。如果你拿着一张纸，试图解出控制声波反射的方程，你不可能解得足够快，来不及看清周围的世界。本·安德伍德居然能在脑子里做这些计算，这真是难以置信。

Our disbelief has a simple explanation, one that stems from the unconscious nature of the mechanisms at work. When someone tells us that Ben Underwood sees the world by analyzing the echoes of his clicks, we imagine him solving complex mathematical equations beyond the reach of a normal human being. That’s both true and false. If you took a piece of paper and tried to solve the equations that govern the reflection of sound waves, you couldn’t do it fast enough to see the world around you. It isn’t believable that Ben Underwood could do the calculations in his head.

没有人能够像学校教的那样，通过有意识的机械方法来解决这些方程式。但是，我们心理可塑性的特殊性在于，通过训练我们的大脑识别出我们意识不到的大量微妙模式，让我们能够无意识地解决问题，而无需陈述问题。

No human is capable of solving these equations in the way that school teaches us to solve equations, by applying a conscious and mechanical method. But the specificity of our mental plasticity is to give us an unconscious means of solving problems without ever stating them, by training our minds to recognize a multitude of subtle patterns that evade our consciousness.

没有数学家会用你在学校学到的方法解决问题。从生物学角度来说，按照这种方法创造真正创新的数学是不可能的，就像从生物学角度来说，通过解牛顿方程来学习走路是不可能的一样。

No mathematician solves problems the way you’re taught in school. It’s biologically impossible to create truly innovative mathematics by following this method, just as it’s biologically impossible to learn how to walk by solving Newton’s equations.

如果不是通过一个你不完全了解的过程，你认为自己是如何学会看东西、走路和说话的？

How do you think you learned how to see, to walk, to talk, if it wasn’t by a process that you weren’t entirely aware of?

如果你用怀疑想象第五维度或通过舌头发出咔嗒声看世界的可能性的标准来评判自己的基本学习经历，你也会得出同样难以置信的结论：理性地讲，学会看、学会走路或学会说话似乎是不可能的。但你却成功做到了。

If you judged your own fundamental learning experiences using the same criteria that make you doubt the possibility of visualizing the fifth dimension or seeing the world by making clicks with your tongue, you would come to the same incredulous conclusion: rationally, learning to see, to walk, or to talk seems impossible. And yet you’ve managed to do it.

2.起点永远是微小的。

2. The starting point is always insignificant.

尝试这样做：闭上眼睛，让某人将手掌直接放在你面前，然后不经意地将手掌拿开再放回原处，同时用舌头发出咔嗒声。你可以听到手掌的存在，就像你在几英寸之内就能听到墙的存在一样。

Try doing this: close your eyes and ask someone to place the palm of their hand directly in front of your face, then take it away and put it back without telling you while you make clicks with your tongue. You can hear the presence of the hand, just as you can hear the presence of a wall if you’re within a few inches of it.

从你已经具备的这种基本能力开始，你可以自由地发展自己的回声定位能力。自己想象这种能力需要某种天赋，但一旦你知道这是可能的，它就不一样了。有人甚至可以教你，比如丹尼尔·基什（他从小就失明，他和本·安德伍德一样，发明了自己的回声定位技术，并把它教给今天的失明年轻人）。

Starting with this primitive ability that you already have in a rudimentary state, you’re free to develop your own ability for echolocation. It takes a kind of genius to dream it up yourself, but it’s not the same thing once you know it’s possible. Someone can even teach it to you, like Daniel Kish (sightless since early childhood, he, like Ben Underwood, invented his own technique of echolocation and teaches it today to young sightless people).

你和其他人一样拥有魔法能力。这只是一个意志、耐心和对世界的开放程度的问题。

You have the same magic abilities as anyone else. It’s just a question of will, patience, and openness to the world.

3.进展缓慢且几乎难以察觉。

3. Progress is slow and almost imperceptible.

心理可塑性本质上是一种缓慢而无形的现象，其进展不可能实时感知。它逐渐改变我们，如此缓慢，以至于一开始我们什么都没有注意到。然而，在某个时刻，我们注意到了，而且通常令人震惊，正是因为我们没有预见到它的到来。它是在不知不觉中发生的，在幕后，我们不需要做任何努力。

Mental plasticity is by nature a slow and invisible phenomenon whose progress is impossible to perceive in real time. It gradually transforms us, so gradually that, at first, we don’t notice anything. At some point, however, we take notice, and it usually comes as a shock, precisely because we didn’t see it coming. It happened unwittingly, in the background, without any effort on our part.

至于回声定位，每天练习一小时，持续两到三周，似乎就能取得显著成果。归根结底，这有点像学习如何开车。

As for echolocation, it seems that you can get significant results by working an hour per day for two to three weeks. In the end, it’s a bit like learning how to drive.

当你想学习一项新运动、一门新语言或一份新工作时，你会经历类似的过程。你必须全身心投入其中，接受自己会感到迷茫，认为自己永远不会擅长这一事实，直到那一刻，你发现，就像被施了魔法一样，你开始掌握它。

When you want to learn a new sport, a new language, or a new job, you go through a similar process. You have to throw yourself into it and accept that you’ll be feeling about for a bit, thinking that you’ll never be any good, until the moment you find, as if by magic, that you’re getting the hang of it.

令人沮丧的完美秘诀

The Perfect Recipe for Discouragement

十几岁的时候，我的表弟杰罗姆买了一块滑板，我感到很震惊：“他为什么会买一块滑板，他又不知道怎么用呢？”他第一次上滑板时就摔倒了。从外面看，学习滑板的人只会把时间都花在从滑板上摔下来上。但过了一会儿，杰罗姆就像变魔术一样，能够做到了。这不仅仅是令人震惊。这变得不公平，令人震惊，好像无能得到了奖励。

As a teenager, when my cousin Jerome bought a skateboard, I was shocked: “Why did he buy a skateboard when he doesn’t know how to use it?” When he first got on it, he fell. Seen from outside, a person learning how to skateboard is just someone who spends all their time falling off it. Except that after a bit, as if by magic, Jerome was able to do it. Then it was more than just shocking. It had become unfair, scandalous, as if incompetence had been rewarded.

只要你忽视心理可塑性的规律，你就会低估别人，也会低估自己。心理可塑性的本质就是把大胆转化为能力。

As long as you ignore the laws of mental plasticity, you underestimate others, and you underestimate yourself. The essence of mental plasticity is to transform audacity into competence.

这个过程是缓慢而无形的，起初似乎无法成功：这就是我们的学习机制的生物学现实。

The process is slow and invisible, and at first success seems unachievable: that’s the biological reality of our learning mechanisms.

不幸的是，这也是让人灰心丧气的完美配方。你需要很强的自控力和自信才能投入到一个令人困惑、缓慢且不确定的过程。

By an unfortunate coincidence, that’s also the perfect recipe for discouragement. You need a lot of self-control and self-confidence to commit to a process that’s confusing, slow, and uncertain.

这就是为什么我们常常局限于学习官方可以学到的东西（有入门或专业发展课程的东西）、可以通过模仿他人学到的东西或自然而然的东西。

That’s why we so often limit ourselves to learning only what’s officially possible to learn (things that have introductory or professional development courses), what you can learn by imitating others, or what comes naturally.

其余的秘密和看不见的学徒被称为“天赋”、“才能”、“超自然力量”。没有人告诉我们，我们可以学会五维视觉，通过回声定位确定方向，或者通过看猫狗的头部来判断它们的性别，所以我们从来没有尝试过。

The rest, the secret and invisible apprenticeships, are said to be “gifts,” “talents,” “supernatural powers.” No one tells us that we can learn to see in five dimensions, get our bearings through echolocation, or tell the sex of dogs and cats by looking at their heads, so we never even try.

我们甚至没有注意到我们在不知不觉中形成的“神奇”力量：从微笑或声音中察觉出虚伪，通过特殊的气味认出我们爱的人，或者在他们说话之前就知道他们会说什么。数学不好的人甚至忘记了，他们在几个小时内掌握的电子游戏在认知上比高中数学课难一百倍。

We go so far as to fail to notice the “magic” powers that we develop without our knowing it: detecting insincerity in a smile or the sound of a voice, recognizing people we love through their particular scent, or knowing what they will say before they even say it. People who aren’t good at math even forget that the video games they master in a matter of hours are cognitively a hundred times more difficult than high school math classes.

重新找回童年时期的学习能力意味着不再相信这些关于天赋和才能的荒唐故事。这意味着再次有能力投入十到二十个小时去做一些可能或不可能的事情，不要因为自己无用而分心。这意味着以开放的心态重新发现世界，尝试某件事只是为了看看会发生什么，为了好玩，因为你想这么做。

Reconnecting with your early childhood capacity for learning means to stop believing in these absurd stories of gifts and talent. It means to become once again capable of devoting ten or twenty hours to something that may or may not be impossible, without being distracted by the feeling of your own uselessness. It means to rediscover the world with an open mind, trying something just to see what happens, for fun, because you want to.

十到二十个小时似乎不算多。通过回声定位观察听起来是个很酷的想法。如果只需要二十个小时，那似乎值得付出这个代价。然而，要花二十个小时做某件事，你必须真的想做。

Ten or twenty hours doesn’t seem like much. Seeing by echolocation sounds like a cool idea. If it only takes twenty hours, it seems worth the price. However, to spend twenty hours at something, you have to really want to do it.

十到二十个小时的真正探索，走出我们的舒适区，足以让我们发现自己身上未知的力量。但最近你有多少次花了十到二十个小时去做一件全新的事情？

Ten or twenty hours of real exploration, outside of our comfort zone, is enough to discover within ourselves unsuspected powers. But how many times lately have you spent ten or twenty hours at something entirely new?

伟大的黑客计划

The Great Hacking Project

当我二十五岁的时候，大约在完成博士学位的前一年，我开始将数学视为一种纯粹的心理重新编程活动，并且我认为我的心理可塑性是没有限制的。

When I was twenty-five, about one year before completing my PhD, I started looking at math as a pure activity of mental reprogramming, and I made the assumption that my mental plasticity had no limits.

或者更坦率地说，我在二十五岁时选择全身心投入到破解我的认知能力的深思熟虑和系统性项目中。

Or, to speak more frankly, I chose at age twenty-five to throw myself into the deliberate and systematic project of hacking my cognitive abilities.

我的基本技巧没有改变：倾听直觉和逻辑之间的矛盾。这项技巧仍然是我探索世界的工具，就像本·安德伍德的舌头一样。

My basic technique hadn’t changed: lending an ear to the dissonance between my intuition and logic. This technique remained my instrument for exploring the world, just like Ben Underwood’s tongue clicks.

在这一刻，我生命中发生了改变，我的信仰体系和随之而来的心态发生了改变。我不再相信我们看待和思考世界的方式是既定事实，我们每个人都有预先设定的智力水平，只能靠自己去适应。取而代之的是，我开始相信我们有自由停止不断地重塑我们的观察和思考方式，日复一日地建构我们自己的智慧。

What changed at this moment of my life were my belief systems and the mindset that ensued. I stopped believing that our way of seeing and thinking about the world was a given fact, and that we each had a predefined amount of intelligence that we had to make do with. In place of that, I began to believe that we had the freedom to ceaselessly refashion our way of seeing and thinking, and to construct our own intelligence day after day.

在第 16 章中，我将讨论一些使我能够遵循这条道路的日益极端的可视化练习。

In chapter 16 I’ll talk about some of the increasingly extreme visualization exercises that allowed me to follow this path.

这种方法的改变最初产生了一个实际结果：我成为了一名富有创造力的数学家。我开始有了以前没有人有过的想法，看到了以前没有人看到过的东西，证明了以前没有人证明过的定理——起初是简单的定理，后来在我的职业生涯中，我证明了那些当时似乎远远超出我能力的定理。数学创造力被认为是科学无法解释的巨大谜团。然而，根据我的经验，一旦我采取了正确的心理态度，它就会自然而然地出现。

This change of approach had one initial practical consequence: I became a creative mathematician. I began to have ideas no one had had before, to see things no one had seen before, to prove theorems no one had yet proven—at first easy theorems, and later in my career theorems that had till then seemed far beyond my capabilities. Mathematical creativity has the reputation of being a great mystery that science can’t explain. In my experience, however, it emerged as a natural phenomenon once I had adopted the correct psychological attitude.

但这种新方法对我的个人生活影响最大。如果我能破解我的视觉皮层并改变我感知空间的方式，如果我能改变我理解真理的方式，那么其他的一切又如何呢？例如，我所相信的我生命中所有既定的事实，这些人们所说的“优点”和“缺点”，构成了我所谓的“个性”，又如何呢？我的害羞、我的心理障碍、我的不安全感，以及所有本应阻碍我前进的东西又如何呢？我的社会身份又如何呢？这些东西怎么会比我对空间和真理的感知更难适应、更难塑造、更难自由地重新编程呢？

But the greatest effect of this new approach was in my personal life. If I was able to hack my visual cortex and modify my way of perceiving space, if I was capable of changing even my way of understanding the notion of truth, what about all the rest? What about, for example, all that I’d believed were givens in my life, these “strengths” and these “weaknesses” that people spoke about and made up my so-called “personality”? What about my shyness, my mental blocks, my insecurities, and everything that was supposed to be holding me back? What about my social identity? How could these things be any less adaptable, less malleable, less freely reprogrammable than my perception of space and truth?

我欣喜地记得这个美丽的日子，当我踏上街头的那一刻，我便说服自己，这些东西是无法修复和确定的，它们必然需要重新配置，而这取决于我的尝试。

I remember with delight this beautiful day when, the very moment I stepped out into the street, I convinced myself that these things couldn’t be fixed and determined, that they were necessarily open to reconfiguration, and that it was up to me to try.

我心想，相信自己具有固定的性格只不过是一种迷信。

Believing that you have a fixed personality, I thought to myself, is nothing but a superstition.

11

球和球棒

11

The Ball and the Bat

一个球和一根球棒总共花费 1.10 美元。球棒比球贵 1 美元。球多少钱？

A ball and a bat cost a total of $1.10. The bat costs $1 more than the ball. How much does the ball cost?

这个问题出自《思考，快与慢》，这是心理学家丹尼尔·卡尼曼的畅销书，他因在认知偏见方面的研究而获得 2002 年诺贝尔经济学奖。

This problem is taken from Thinking, Fast and Slow, the best-selling book by the psychologist Daniel Kahneman, winner of the 2002 Nobel Prize in Economics for his work on cognitive biases.

我鼓励你和朋友一起做这个测试。它几乎总是有效的：大多数人回答说球的价格是 10¢。但这不是正确答案。如果球的价格是 10¢，那么球棒的价格将是 1.10 美元（因为它比球贵 1 美元），而球和球棒加起来的价格是 1.20 美元。

I encourage you to try the test with your friends. It works nearly all the time: most people answer that the ball costs 10¢. But that’s not the right answer. If the ball cost 10¢, the bat would cost $1.10 (since it costs $1 more than the ball), and the ball and the bat together would cost $1.20.

如果你解释为什么他们的答案是错误的，你的朋友很容易就会明白。但他们不一定知道正确答案。他们甚至会找出很多借口：计算很难，他们必须写下方程式，但他们没有笔，他们懒得写……

If you explain why their answer is wrong, your friends will get it easily enough. But they won’t necessarily know the right answer. They’ll even find a lot of excuses: it’s hard to do the calculations, they’d have to write down the equations but they don’t have a pen, they can’t be bothered. . . .

正确答案是 5¢。如果球的价格为 5¢，球棒的价格为 1.05 美元，两者合计价格为 1.10 美元。

The correct answer is 5¢. If the ball costs 5¢, the bat costs $1.05, and together they cost $1.10.

球和球棒问题在卡尼曼的书中占据着重要地位，因为它完美地诠释了他的理论。根据他的说法，我们有两个不同的认知系统，他称之为系统 1和系统 2。

The ball and bat problem plays a prominent role in Kahneman’s book because it’s the perfect illustration of his theory. According to him, we have two distinct cognitive systems, which he calls System 1 and System 2.

系统 1 让你无需费力就能做出即时而本能的反应。当有人问你 2 + 2 等于多少时，你出生在哪一年，大象和老鼠哪个更重，你甚至不用思考。但系统 1 也会让你错误地回答，这个球值 10 美分。

System 1 allows you to give immediate and instinctive responses, without even trying. When someone asks you how much is 2 + 2, what year you were born, which weighs more, an elephant or a mouse, you don’t even have to think. But it’s also System 1 that makes you answer, incorrectly, that the ball costs 10¢.

当你被要求计算 47 × 83 或从你出生到现在过了多少天时，你必须使用系统 2。你知道如何得到答案，但你必须思考。你可能需要铅笔和纸。有一件事是肯定的：你真的不想这样做。即使系统 2 更可靠、更严格，你也只有在别无选择时才会使用它，因为认真思考、计算和逻辑推理都很累人。

System 2 is what you have to use when you’re asked to calculate 47 × 83, or how many days have passed since your birth. You know how to get the answer, but you’d have to think. You probably need pencil and paper. One thing is certain: you don’t really want to do it. Even if System 2 is more reliable and rigorous, you only use it when you have no other choice, because thinking hard, doing calculations, and logical reasoning are all tiresome.

卡尼曼的理论可以概括如下：

Kahneman’s theory can be summed up as follows:

1. 每当我们的系统 1 给出答案时，我们都会忍不住直接使用它，而不调用系统 2，甚至不去验证答案是否正确。由于系统 2 消耗大量的脑力和资源，我们主要依靠直觉。从生物学角度来看，我们已经形成了对智力懒惰的偏好。

1. Each time our System 1 gives us an answer, we’re tempted to use it without calling on System 2, not even to verify that the answer is correct. Because System 2 uses a lot of mental energy and resources, we primarily rely on our instinct. Biologically, we’ve developed a preference for intellectual laziness.

2. 在某些情况下，我们的系统 1 是系统性错误的。我们总是会犯同样的错误，就好像我们大脑中的接线图有缺陷一样。这些就是卡尼曼和他的学派着手研究的“认知偏差”。我们都想说这个球的价格是 10 美分。

2. In certain situations, our System 1 is systematically wrong. We all make the same mistakes, all the time, as if the wiring schematic in our brain was defective. These are the “cognitive biases” that Kahneman and his school have set out to study. We all want to say that the ball costs 10¢.

卡尼曼的书之所以成为畅销书，部分原因是它超越了简单的理论观察，并提出了避免落入陷阱的具体方法。

Kahneman’s book became a best seller in part because it went beyond the simple theoretical observation and proposed a concrete method to avoid falling into the trap.

他提出了一个简单的建议：熟记书中列出的认知偏差列表，每当你认识到一种典型情况时，就要克服自己的倾向，使用你的系统 2，同时试着忽略你的系统 1。

He has a simple recommendation: learn the list of cognitive biases presented in his book by heart, and each time you recognize one of the typical situations, fight your inclination and use your System 2 while trying to ignore your System 1.

我认为我有更好的方法，我会解释的。

I think I have a better way of doing it, which I’ll explain.

“这是作弊！”

“That’s cheating!”

我第一次听到球和球棒的故事，是从一个在普林斯顿大学学习认知科学的朋友那里听说的。她刚读完卡尼曼的书，想和我一起做测试。

The first time I heard the story of the ball and the bat, it was from a friend who was studying cognitive science at Princeton. She had just read Kahneman’s book and wanted to do the test with me.

和大多数人一样，我本能地回答。我听了我的系统 1，却不知道它叫系统 1。没有思考，没有做任何计算，我给出了第一个浮现在脑海中的答案：“5 美分。”

Like most people, I gave an instinctive response. I listened to my System 1 without knowing it was called System 1. Without thinking, without doing any calculations, I gave the first answer that popped into my head: “5¢.”

我觉得我的回答惹恼了我的朋友，但我当时不知道为什么。她花了时间解释了发生了什么。我应该回答“10¢”，或者至少花几秒钟再回答“5¢”。无论如何，我不可能不花点时间思考就立即回答“5¢”。这根本是不允许的。甚至有一个人因为证明这是不可能的而获得了诺贝尔奖。

I felt that my answer annoyed my friend but I didn’t immediately know why. She took the time to explain what was up. I was supposed to answer “10¢,” or at least take a few seconds before answering “5¢.” At any rate, there was no way I was supposed to answer “5¢” immediately, without taking the time to think about it. That was simply not allowed. A guy had even won the Nobel Prize for showing it was impossible.

很快，就在转移话题之前，我的朋友给出了一个解释——一个简单、务实且并非完全错误的解释：“那是作弊！你是个数学家！”

Quickly enough, right before changing the topic of conversation, my friend did however come up with an explanation—a simple, pragmatic, and not entirely false one: “That’s cheating! You’re a mathematician!”

当我让我的朋友和同事参加测试时，我真的很惊讶地发现他们中有这么多人回答“10¢”，更让我惊讶的是，在承认他们最初的回答是错误的之后，他们很难找到正确的答案。最不可思议的是，每个人都跟我谈论“做计算”，好像正确答案是“5¢”并不明显。

When I had my friends and colleagues take the test, I was sincerely surprised to find that so many of them answered “10¢,” and even more surprised at their difficulty in finding the right answer after admitting that their initial response was wrong. The most incredible thing was that everyone spoke to me about “doing the calculations,” as if it weren’t visually evident that the right answer was “5¢.”

我就像道尔顿和他的魔法天竺葵一样，只不过我不是缺少视锥细胞的人，而是比我的朋友看到​​的颜色更多。当然，我和道尔顿的另一个不同之处在于，这种解释与遗传学无关。

I was like Dalton with his magic geranium, except that instead of being the one with the missing cones, I was the one seeing more colors than my friends. The other difference with Dalton, of course, is that the explanation doesn’t have anything to do with genetics.

在本章的最后，我将解释我做了什么才找到正确答案——以及您如何学习这样做。

At the end of this chapter, I’ll explain what I did to see the right answer—and how you can learn to do it as well.

A 或 B

A or B

这个有关球和球棒的故事真的开始引起我的兴趣，我试图理解是什么阻止我的朋友们看到如此明显的正确答案。

This story about the ball and the bat really began to intrigue me, and I tried to understand what was stopping my friends from seeing the right answer when it was so obvious.

和道尔顿一样，我开始了小小的探索。我相信我找到了一个解释。在向朋友询问了球的价格后，我又问了这个问题：

A bit like Dalton, I began my little inquiry. I believe I’ve found an explanation. After asking my friends about the price of the ball, I followed up with this question:

想象一下，你必须在生活中做出一个重要的决定。你可以在选项 A 和选项 B 之间做出选择。你的直觉告诉你选择 A，但你的理智告诉你选择 B。你会怎么做？

Imagine that you have to make an important decision in your life. You have the choice between option A and option B. Your intuition tells you to choose A, but your reason tells you to choose B. What do you do?

我向十几位非数学家朋友提出了这个问题，几乎所有人都毫不犹豫地回答说，他们会遵循直觉选择 A。只有一个人选择了 B。另一个人犹豫了很久，没有给出明确的答案。

I presented this question to more than a dozen of my nonmathematician friends, and almost all of them answered, without hesitation, that they’d follow their intuition and choose A. Only one person chose B. Another hesitated for a long time, without ever giving a clear response.

如果你自己尝试这个实验，没有什么可以保证你会得到如此高比例的人选择 A。我的协议受到所谓的选择偏差的影响，因为我的朋友不一定代表一般人群，而且很可能听从直觉的人更有可能成为我的朋友。

Nothing guarantees that you’d get such a high percentage of people choosing A if you tried this experiment on your own. My protocol suffers from what is called selection bias, in that my friends aren’t necessarily representative of the general population, and it may very well be that people who listen to their intuition have greater chances of becoming my friends.

我其实并不关心 A 和 B 的具体比例。我想知道是否有人会给出和我一样的答案。没有人能做到。

The exact proportion of A and B didn’t really interest me. What I wanted to know was whether someone would come up with the same response I would have given myself. No one did.

我的假设是，我对这个问题的不同寻常的回答是让我擅长数学的关键，并且在此过程中，重新教育了我的许多认知偏见。

My hypothesis is that my unusual response to the question is the key that allowed me to become good at math and, along the way, to reeducate many of my cognitive biases.

不合理的假设

An Unreasonable Assumption

卡尼曼说，数千名美国学生参加了球棒测试，结果“令人震惊”。在二流大学，错误率超过 80%。甚至哈佛大学、麻省理工学院和普林斯顿大学的学生答错率也超过 50%。

Kahneman says that thousands of American students took the ball and bat test, and that “the results are shocking.” At the lower-tier universities, the error rate was over 80 percent. Even students at Harvard, MIT, and Princeton gave the wrong answer more than 50 percent of the time.

卡尼曼的书很吸引人，但每当我看到他将“正确答案”与“直觉答案”对立起来时，我都会感到困惑，好像直觉答案只有一个可能，而且它一定是错误的。例如，他写道：“可以肯定的是，直觉答案也出现在那些最终得出正确数字的人的脑海中——他们不知何故设法抵制了直觉。”

Kahneman’s book is fascinating, but I’m confused whenever I see him opposing “the right answer” and “the intuitive answer,” as if there were only one intuitive response possible, and it was necessarily false. For example, he writes: “It is safe to assume that the intuitive answer also came to the mind of those who ended up with the correct number—they somehow managed to resist the intuition.”

换句话说，卡尼曼认为我不应该存在是安全的。我的观点是，这可以理解，这不是一个合理的假设。

In other words, Kahneman finds it safe to assume that I shouldn’t exist. My opinion, understandably, is that this isn’t a reasonable assumption.

但除了我自己的存在这个相对次要的问题之外，这件轶事揭示了卡尼曼的理论与所有数学家内心深处的认知之间的重大差距。谁最有资格给你提供心算方面的建议，这取决于你。

But beyond the relatively minor question of my own existence, this anecdote reveals a major disconnect between Kahneman’s theory and what all mathematicians know deep in their bones. It’s up to you to decide who’s in the best position to give you advice on mental calculations.

卡尼曼发现哈佛大学、麻省理工学院和普林斯顿大学 50% 的学生盲目依赖明显错误的直觉，这让他感到震惊，我和他一样震惊。

Kahneman finds it shocking that 50 percent of the students at Harvard, MIT, and Princeton blindly relied on a manifestly false intuition, and I’m as shocked as he is.

但我同样对卡尼曼似乎认为完全正常的事情感到震惊：为什么哈佛大学、麻省理工学院和普林斯顿大学有 50％ 的学生在直觉如此错误的情况下仍然被录取？

But I’m equally shocked by something that Kahneman apparently finds completely normal: how is it that 50 percent of the students at Harvard, MIT, and Princeton managed to get accepted despite having such faulty intuitions?

我在竞争激烈的大学学习和教学，我知道那些能在脑海中直接“看到”正确答案的学生具有巨大的竞争优势。我不明白其他人怎么能与之竞争。我猜他们用以下方法弥补密集的复习，这是我完全没有能力做到的，一想到这件事我就头疼。

Having studied and taught at highly competitive universities, I know that students who can directly “see” the correct answer in their head have an enormous competitive advantage. I don’t understand how the others can even compete. I imagine that they compensate by intensive cramming, something I’m completely incapable of and the very thought of which gives me a headache.

卡尼曼的建议包括确定我们应该“抵制”直觉并服从系统 2 的情况。这个建议很奇怪，因为它出自一个一生都在记录我们对努力的厌恶、对本能和即时反应的偏好、对系统 1 的过度热爱——以及对系统 2 的憎恨的人之口。

Kahneman’s advice consists of identifying the situations where we should “resist” our intuition and submit ourselves to System 2. It’s strange advice coming from someone who’s spent his life documenting our aversion to effort, our preference for instinctive and immediate responses, our immoderate love for System 1—and our hatred of System 2.

这种认为我们应该抵制不良本能并完全服从机械思维模式的想法曾经是教育界的主流范式。卡尼曼很清楚为什么这种模式行不通。

This idea that we should resist our bad instincts and entirely submit to a robotic mode of thought was once the prevalent paradigm in education. Kahneman is well enough placed to know why it can’t work.

还有一点让我很困惑。我们确实应该警惕我们的系统 1。但是我们该如何对待我们的系统 2？就我个人而言，九年级之后，当我发现自己无法连续计算三行而不犯错误时，我就不再信任我的系统了。

Another point bothers me. It’s true that we should be wary of our System 1. But what are we to make of our System 2? Personally, I stopped trusting mine after ninth grade, when I found out I wasn’t able to string together three lines of calculations without making a mistake.

但最令人不安的是，卡尼曼的推理方式好像我们的直觉是天生的，我们不可能对其进行重新配置或重新编程。如果他生活在古罗马时代，他几乎肯定会说，不可能在心里表示出“1,000,000,000 - 1”这个运算的结果，因为这个数字大大超出了人类直觉的能力。

But the most troubling aspect is that Kahneman reasons as if our intuition were hardwired, with no possibility for us to reconfigure or reprogram it. Had he lived in the ancient Roman era, he would almost certainly have said that it was impossible to represent mentally the result of the operation “1,000,000,000 – 1,” because the number greatly exceeded the capacities of human intuition.

系统 3

System 3

当我需要在生活中做出一个重要的决定时，如果我的直觉告诉我选择选项 A，而我的理智告诉我选择选项 B，我会告诉自己有些事情正在发生，而我还没有准备好做出决定。

When I need to make an important decision in my life, if my intuition tells me to choose option A and my reason tells me to choose option B, I tell myself there’s something going on and I’m not ready to make the decision.

这时候就需要采用我所说的系统 3。

That’s the moment to resort to what I call System 3.

系统 3 是一系列内省和冥想技巧，旨在建立直觉与理性之间的对话。每次你试图回忆梦境时，你都会用到它，用语言描述那些让你感到奇怪的瞬间印象，理清你最困惑和最矛盾的想法。

System 3 is an assortment of introspection and meditation techniques aimed at establishing a dialogue between intuition and rationality. You use it each time you try to recall your dreams, to put words to the fleeting impression that left a strange taste in your mouth, to sort out your most confused and contradictory ideas.

当我十八岁的时候，我发现，只要我努力描述和命名，我脑海中那些愚蠢的形象就会自我纠正，当我养成了倾听直觉和逻辑之间不一致的习惯后，我把系统 3 放在了我学习数学的策略的核心位置。结果超出了我的最大预期。

When I was eighteen and I discovered that the stupid images in my head had a tendency to correct themselves once I made the effort to describe and name them, when I got into the habit of lending an ear to the dissonance between my intuition and logic, I put System 3 at the center of my strategy for learning math. The results exceeded my wildest expectations.

我们都知道系统 3，也都使用它，至少时不时会用。我的数学之旅让我明白，自愿和彻底地使用系统 3 不仅是可能的，而且它还能增强我们的直觉能力，远远超出人类认知的所谓极限。

We all know System 3 and we all use it, at least from time to time. My mathematical journey taught me that a voluntary and radical use of System 3 is not only possible, it augments our intuitive capacities well beyond the supposed limits of human cognition.

多年来，系统地寻求直觉和逻辑之间更好的结合已经成为我理解世界、他人甚至我自己的方式。

Through the years, the systematic search for a better alignment between my intuition and logic has become my way of understanding the world, others, and even myself.

从实际角度来说，这意味着：当我的直觉告诉我 A 而理性告诉我 B 时，我会把自己置于裁判的位置。我强迫自己将直觉转化为文字，像讲一个简单易懂的故事一样讲述它。反之亦然，我试图想象逻辑推理实际上在表达什么，用我的身体去体验它，去听它想要表达什么。我问自己是否真的相信它。我摸索着。这需要时间，但并不是真正的努力。这更像是对流水的冥想，背景中发生的事情可能会停止和开始，然后在几天、几个月甚至几年后突然变得清晰起来。

In practical terms, here’s what that means. When my intuition tells me A and rationality tells me B, I put myself in the position of a referee. I force myself to translate my intuition into words, to tell it like a simple and intelligible story. Vice versa, I try to picture what logical reasoning is actually expressing, to experience it in my body, to hear what it’s trying to say. I ask myself if I really believe it. I fumble about. It takes time but it’s not a real effort. It’s more like a meditation on running water, something going on in the background that might stop and start, then all of a sudden become clear days, months, or even years later.

目标是了解哪里出了问题。我的直觉和逻辑是否在讲同一种语言？它们是否在谈论同一件事？

The goal is to understand where things are going wrong. Are my intuition and logic even speaking the same language? Are they even talking about the same things?

我的直觉从来都不是完美的。它通常很准确，但有时却很荒谬。好消息是它通常可以修正。至于逻辑，它从来都没错。至少在官方意义上。只是它不一定能说出我的想法。

My intuition is never perfect. It’s often relevant, but sometimes it’s just rubbish. The good news is that it’s generally fixable. As for logic, that’s never wrong. At least officially. Except that it doesn’t necessarily say what I think it’s saying.

最终，几乎总是我的直觉获胜。当我强迫它听从逻辑时，它会考虑到这一点并调整自己的立场。逻辑是惰性的，就像鹅卵石一样。我的直觉是有机的，它是活的，而且在成长。

In the end, it’s almost always my intuition that wins. When I force it to listen to what logic is saying, it takes that into account and adjusts its position. Logic is something inert, like a pebble. My intuition is organic, it is living and growing.

显然，将这​​种方法称为系统 3是愚蠢的。它应该简单地称为思考或反思。但这些词的含义已被一种传统所劫持，这种传统想让我们相信，我们应该违背直觉去思考。我们被告知，直觉是理性的死敌，两者之间的任何对话都是不可能的，而思考意味着你必须盲目地服从系统 2。

It’s obviously stupid to call this approach System 3. It should simply be called thinking or reflecting. But the meaning of these words has been hijacked by a tradition that wants to make us believe that we should think contrary to our intuition. We’re told that our intuition is the mortal enemy of reason, that any dialogue between the two is impossible, and thinking means you have to submit blindly to System 2.

我个人无法违背直觉去思考，并且我严重怀疑那些声称可以这样做的人的诚意。

I’m personally incapable of thinking against my intuition and I have serious doubts as to the sincerity of people who claim they can.

在第三章中，我说过直觉是你最强大的智力资源。然而，尽管可能会破坏你的梦想，我必须坦诚地告诉你：你的直觉不是灵丹妙药，也不是你的幸运星，也不是上帝之手。它比这些要琐碎得多。它是看不见却完全具体和物质化的现实的有形表现：你的神经元之间的突触连接纠缠，你的大脑不断构建和重组，就像你在子宫里一样。

In chapter 3 I said that your intuition was your most powerful intellectual resource. However, at the risk of spoiling your dreams, I must be honest with you: your intuition isn’t a magical elixir, or your lucky star, or the hand of God on your shoulder. It’s much more trivial than that. It’s the tangible manifestation of a reality that is invisible but perfectly concrete and material: the entanglement of synaptic connections between your neurons that your brain continuously constructs and reorganizes, as it has done since you were in the womb.

你的大脑包含的神经元数量与银河系中的恒星数量一样多。平均而言，每个神经元都与数千个其他神经元相连。这种由数百万亿个互连组成的结构就是你的心理联想网络。它的结构是你赋予不断涌入的原始信息意义的方式进入你的大脑。这实际上是你对世界的看法。你所看到、听到、感受到、想象或渴望的一切，你所有的经历、你所知道的一切、你所记得的一切，都被编码在这个网络中。当你的直觉说话时，它就是从这里说话的。

Your brain contains as many neurons as there are stars in the Milky Way. Each of these neurons is, on average, tied to thousands of other neurons. This fabric of hundreds of trillions of interconnections is the network of your mental associations. Its structure is your way of giving meaning to the raw information continually flooding into your brain. This is, literally, your vision of the world. All that you have seen, heard, felt, imagined, or desired, all of your experience, all that you know, all that you remember, is encoded in this web. When your intuition speaks, that’s where it’s speaking from.

你的直觉总是比最复杂的语言推理更强大、更明智。尽管如此，直觉也不是绝对可靠的。如果你的直觉告诉你球的价格是 10 美分，那它显然是错的。

Your intuition will always be more powerful and better informed than the most sophisticated of language-based reasonings. For all that, it’s not infallible. If your intuition tells you the ball costs 10¢, it’s plainly wrong.

我的直觉并不比你的直觉更不可靠。它总是出错。然而，我已经学会了永远不要为此感到羞耻。我不会鄙视我的错误，也不会把它们放在一边，因为我不认为它们暴露了我的智力低下或我大脑中根深蒂固的认知偏见。恰恰相反。没有什么比一个明显的错误更令人兴奋的了：它总是表明我没有以正确的方式看待事物，并且有可能更清楚地看待它们。当我能够指出直觉中的错误时，我知道这是个好消息，因为这意味着我的心理表征已经在重新配置自己了。

My intuition isn’t any less fallible than yours. It’s always getting things wrong. I have, however, learned never to be ashamed of it. I don’t disdain my mistakes, I don’t push them aside, because I don’t think that they betray my intellectual inferiority or some cognitive biases hardwired in my brain. On the contrary. Nothing’s more exciting than a big glaring error: it’s always a sign that I’m not looking at things in the right way, and that it’s possible to see them more clearly. When I’m able to put my finger on an error in my intuition, I know it’s good news, because it means that my mental representations are already in the process of reconfiguring themselves.

我的直觉只有两岁孩子的心智年龄——它没有抑制，总是想要学习。如果你不再虐待自己的直觉，你会发现它和我的直觉一模一样，只要求被允许成长。

My intuition has the mental age of a two-year-old—it has no inhibitions and always wants to learn. If you stop mistreating your own, you’ll see that it’s exactly like mine, only asking to be allowed to grow.

球的代价

The Price of a Ball

因为我的字写得很糟糕，而且我很容易分心，所以我很容易在计算中犯错误。

Because I have terrible handwriting, and because I’m easily distracted, I have a tendency to make mistakes in calculations.

我在九年级时发现，解决这个问题的唯一方法是每写三行就验证一下我写的内容是否有意义，我是否真的相信它。换句话说，我学会了如何使用我的系统 1 来监督我的系统 2 的工作。从那时起，我就无法操纵我没有直觉的数学对象了。

I discovered in ninth grade that the only way to get around that was to verify after every three lines that what I was writing still made sense and that I really believed it. In other words, I learned how to use my System 1 to supervise the work of my System 2. From this time on, I was incapable of manipulating mathematical objects that I had no intuition for.

什么时候我不再主要通过书写形式来形象化数字了？我不记得了。但毫无疑问可以追溯到同一时期。十进制数字对于书面计算很有用，但当你想对这些计算的有效性形成直观的想法时，它肯定不那么实用。这就是系统 1 的优势所在：它不受语言和书写的限制。

At what moment did I stop primarily visualizing numbers through their written form? I don’t recall. But it undoubtedly goes back to the same period. Decimal writing of numbers is useful for written calculations but it is certainly less practical when you want to form an intuitive idea about the validity of those calculations. This is where System 1 has an edge: it isn’t bound by the constraints of language and writing.

根据具体情况，我有许多不同的方式来形象化数字。例如，我倾向于用长度来形象化价格。当我的朋友告诉我球和球棒加起来的价格是 1.10 美元时，我立即将她的话转化为如下的心理图像：

Depending on the context, I have many different ways of visualizing numbers. I have, for example, a tendency to visualize price in terms of length. When my friend told me that the ball and bat together cost $1.10, I immediately translated her words into a mental image that looked something like this:

当她告诉我球棒比球贵 1 美元时，我是这样看的：

When she told me that the bat cost $1 more than the ball, here’s how I saw it:

然后这两个图像在我脑海里融合在一起并变成了这样：

Then the two images came together in my head and morphed into something like this:

如果您这样想象这个问题，那么不需要天才就能算出一个球的价格是 5¢。

If this is how you visualize the problem, it doesn’t take a genius to figure that a ball costs 5¢.

心理意象本身无所谓好坏。它的价值在于它能让你理解什么。有无数种方法可以将问题形象化，我并不认为我的方法更好。我的数字直觉并不那么出色。如果球和球棒的总价为 2,734.18 美元，而球棒比球贵 967.37 美元，我就无法计算了。

A mental image is neither good nor bad in and of itself. Its value lies in what it allows you to understand. There are countless ways to visualize the problem and I don’t pretend that mine is better. My numeric intuition isn’t all that remarkable. I wouldn’t be able to do the calculation if the ball and bat together cost $2,734.18 and the bat cost $967.37 more than the ball.

我脑子里有这些画面，因为在我的生活中，我犯过很多计算错误。我没有得出结论说我数学不好，而是寻找更简单的方式来看待事物，理解我所写的内容。

I have these pictures in my head because, in my life, I’ve made a lot of calculations errors. Instead of concluding that I was terrible at math, I simply looked for simpler ways to see things, to grasp what I was writing.

随着时间的推移，通过这种方法，我构建了各种各样的心理图像，帮助我今天更好地理解世界。

In time, with this approach, I constructed a great variety of mental images that help me today to better understand the world.

如果你想学会如何显而易见地发现球的价格是 5¢，我建议你像数学家一样面对新的、难以理解的想法。与其死记硬背我的图，不如训练自己构造适合你的图。最重要的信息，你应该永远记住的信息，是：

If you want to learn to find it obvious that the ball costs 5¢, I recommend proceeding like a mathematician would when faced with a new and incomprehensible idea. Rather than learning my pictures by heart, train yourself to construct pictures that work for you. The most important messages, the ones you should always bear in mind, are these:

1.你可以重新编程你的直觉。

1. You can reprogram your intuition.

2. 直觉和理性之间的任何不一致都是你在内心创造新视角看待事物的机会。

2. Any misalignment between your intuition and reason is an opportunity to create within yourself a new way of seeing things.

3. 不要指望一切都会实时地发生。形成心理意象意味着重新组织神经元之间的连接。这个过程是有机的，有自己的节奏。

3. Don’t expect it all to come at once, in real time. Developing mental images means reorganizing the connections between your neurons. This process is organic and has its own pace.

4. 不要强迫自己。只需从你已经理解的、你已经看到的、你觉得容易的开始，然后玩一玩。试着直观地解释你写下的计算。如果有帮助的话，在纸上写写画画。

4. Don’t force it. Simply start from what you already understand, what you can already see, what you find easy, and just play with it. Try to intuitively interpret the calculations you would have written down. If it helps, scribble on a piece of paper.

5.随着时间和练习，这项活动将增强你的直觉能力。你可能看起来并没有取得任何进步，直到有一天，正确答案突然变得显而易见。

5. With time and practice, this activity will strengthen your intuitive capacities. It may not seem like you’re making progress, until the day the right answer suddenly seems obvious.

你需要进行多次训练。具体要多少次，我不知道。不值得让自己筋疲力尽——最好将其分成五分钟的短时间，在洗澡或散步时思考。最重要的是，慢慢来。最好每周或每月只考虑一次。最重要的是，坚持下去，不要放弃。最终会实现的。

You’ll need a number of training sessions. Exactly how many, I don’t know. It’s not worth tiring yourself out—better to split it up into short five-minute sessions, and think about it in the shower or while on a walk. Above all, take your time. It’s good to think about it only once a week or once a month. Most important, keep at it and don’t let it drop. It will come eventually.

解决问题永远只是借口。重要的是你有能力重新培养你的直觉，获得对你的身体和思想的信心。

Solving a problem is only ever a pretext. The important thing is that you have the power to reeducate your intuition, to gain confidence in your body and thoughts.

对此你不会感到惊讶。解决球和球棒的问题就像站在冲浪板上一样。卡尼曼说，第一次站在冲浪板上时，你会掉进水里，并得出结论，人类天生就有平衡感缺陷，站在冲浪板上永远不可能成为直觉。他的建议是离开水面，用心学习物理定律。我的建议是重新站在冲浪板上。

Nothing about this should surprise you. Solving the problem of the ball and the bat is like standing up on a surfboard. Kahneman says that the first time you stand up on a surfboard, you’ll fall in the water, and concludes that humans are born with a defective sense of balance and that getting up on a surfboard can never become intuitive. His advice is to get out of the water and learn the laws of physics by heart. My advice is to get back up on the board.

电气、机械、有机思维

Electrical, Mechanical, Organic Thought

本书的中心思想是，我们的文化传达了关于大脑如何运作的错误信念，而这些错误的信念使人们远离了可以让他们擅长数学的简单行动。

A central idea of this book is that our culture conveys false beliefs about how our brain functions, and that these false beliefs keep people away from the simple actions that would allow them to become good at math.

当你告诉人们某些真理本质上是违反直觉的，你是在告诉他们，他们永远无法真正理解。这是一种让他们气馁的方式。没有什么是本质上违反直觉的：有些东西只是暂时违反直觉的，直到你找到让它变得直观的方法。

When you say to people that certain truths are by nature counterintuitive, you tell them that they can never really understand. It’s a way of discouraging them. Nothing is counterintuitive by nature: something is only ever counterintuitive temporarily, until you’ve found means to make it intuitive.

理解某件事就是让自己直观地了解它。向别人解释某件事就是提出简单的方法让其直观地了解它。

Understanding something is making it intuitive for yourself. Explaining something to others is proposing simple ways of making it intuitive.

这些都不会削弱卡尼曼研究的价值。他记录的认知偏见是具有重大社会意义的惊人人类现实。我们都有偏见，即使这些偏见不是固定的，每个人的情况也不同，而且某些偏见比其他偏见更为普遍和成问题。

None of this takes away from the value of Kahneman’s work. The cognitive biases that he documents are striking human realities of great social importance. We all have biases, even if they aren’t hard-coded and vary from one person to another, and certain biases happen to be more widespread and problematic than others.

卡尼曼对系统 1 和系统 2 的区分具有简单的优点。从某种意义上说，它继承了左脑和右脑之间的经典对立，但采用了现代版本，没有解剖学上的胡言乱语。这只是一个基本模型，但它很有吸引力，并帮助我们意识到我们调动心理资源的不同方式。

The distinction Kahneman makes between System 1 and System 2 has the merit of being simple. In a sense, it picks up the classic opposition between left brain and right brain, but in a modern version, without the anatomical nonsense. It’s just a basic model, but it’s appealing, and helps us become aware of our different ways of mobilizing our mental resources.

我们将以系统 3 的典型原理总结结束本章，这是卡尼曼理论的一大疏忽。我们将在第 19 章中再次讨论我们大脑皮层的物理结构及其功能，这将使我们能够从生物学角度解释系统 3 及其功效。无论如何，系统 3 是数学工作真实性质的一个很好的模型。

We’ll end this chapter with a summary of the characteristic principles of System 3, the big oversight in Kahneman’s theory. We’ll talk again in chapter 19 about the physical structure of our cortex and how it functions, which will allow for a biological interpretation of System 3 and its efficacy. At any rate, System 3 is a good model for the real nature of mathematical work.

系统 1 是我们的直觉能力。我们都喜欢用电的比喻来描述它：我们说直觉让我们的思考速度快如闪电。这并非完全错误。我们的大脑严格来说不是一个电路，但沿着神经元传输的信号本质上是电的。

System 1 is our intuitive capacity. We all like to describe it using electrical metaphors: with our intuition, we say that we think with the speed of lightning. It’s not entirely false. Our brain isn’t properly speaking an electrical circuit, but the signal that’s transmitted along the neurons is electrical in nature.

系统 2 是我们进行严格推理的能力。我们用机械术语来想象它，比如齿轮或类似的东西。这与任何生物现实都不相符。从生物学角度来看，我们能够假装自己是机器人，机械地应用一系列预设的指令。有了正确的指令，我们可以得出合乎逻辑的结论和有效的计算。但这是如此令人讨厌和违背我们的天性，以至于我们通常会在几秒钟或最多几分钟后放弃。最终，我们很遗憾机器人：我们犯了太多错误，无法坚持到底。

System 2 is our capacity for rigorous reasoning. We imagine it in mechanical terms, with gears or something of the sort. That doesn’t correspond to any biological reality. What we are biologically capable of is to pretend we’re robots and mechanically apply a preset series of instructions. With the right set of instructions, we can make logical conclusions and valid calculations. But it’s so disagreeable and against our nature that we usually give up after a few seconds, or at most a few minutes. In the end we’re rather sorry robots: we make too many mistakes and we can’t go the distance.

快速思考、慢思考和超慢思考

Thinking fast, slow, and super slow

我们的文化完全忽视了系统 3，我找不到合适的词来描述它。正如我上面所说，我想说系统 3 只是对应于我们的思考能力。但动词“思考”并没有多大意义，因为它已被用作服从系统 2 的命令。

System 3 is so entirely ignored by our culture that I can’t find the right word to characterize it. As I said above, I would like to say that System 3 simply corresponds to our capacity for thinking. But the verb to think doesn’t mean much since it’s been used as an injunction to submit to System 2.

系统 3 的活动是一种特殊的冥想，但这个词也太模糊了。并非所有的冥想都是系统 3 的活动。系统 3 的具体目标是在系统 1 和系统 2 之间建立对话，以了解它们的不一致之处并加以解决。它不是自由的冥想，而是受非矛盾原则约束的冥想。它的最终目标是在考虑系统 2 的结果的同时修改和更新系统 1。

The activity of System 3 is a special kind of meditation, but this word is also much too vague. Not all meditation is an activity of System 3. System 3 specifically aims at establishing a dialogue between Systems 1 and 2, in order to understand their misalignments and resolve them. Rather than a free meditation, it’s one constrained by the principle of noncontradiction. Its ultimate goal is to revise and update System 1 while taking into account the results of System 2.

还必须将系统 3 与我们的系统 1 可以自行修改，而无需我们采取任何刻意行动。我们的心理可塑性源于突触网络的不断重新配置：我们的心理回路会随着我们的经历而发展。你可以把神经元想象成微小的植物，它们会生长，并将根扎得越来越深。

It’s also necessary to distinguish System 3 from the capacity of our System 1 to revise itself without any deliberate action on our part. Our mental plasticity results from the constant reconfiguration of our synaptic network: our mental circuits evolve in response to our experiences. You can imagine neurons as minuscule plants that grow and sink their roots deeper and deeper.

每次我们练习一项活动时，我们都会让我们的系统 1 适应这项活动的具体细节。当我们试图站在冲浪板上时，我们会让系统 1 适应牛顿物理学的严酷现实，并培养我们的冲浪本能。通过系统 3，我们会让系统 1 适应逻辑一致性的严酷现实，并培养我们对真理的本能。

Every time we practice a given activity we habituate our System 1 to the specifics of that activity. When we try to stand up on a surfboard, we habituate our System 1 to the hard realities of Newtonian physics and construct our surfing instincts. With System 3, we habituate our System 1 to the hard realities of logical consistency and construct our instincts for truth.

对数学教学的巨大误解源于这样一个事实：数学的所有有形表现——令人困惑的语言、难以理解的符号、怪异而僵化的推理——似乎都将其与系统 2 联系在一起。

The great misunderstanding of math teaching stems from the fact that all the tangible manifestations of mathematics—its confusing language, its incomprehensible notation, its bizarre and rigid reasoning—seem to tie it to System 2.

大多数人都对此深信不疑。几分钟后他们就会灰心丧气，或者开始自虐，但成功的可能性为零。

Most people take that at face value. They become discouraged in a few minutes, or throw themselves into a masochistic effort that has zero chance of succeeding.

但有些人选择依赖他们的系统 3。他们没有意识到自己在做什么特别的事情。数学对他们来说很简单。它甚至不觉得是在工作。他们只是在脑海中看到画面，每天花几分钟看这些画面并问自己一些幼稚的问题。

But a few people choose to rely on their System 3. They’re not aware that they’re doing anything special. Math just feels easy to them. It doesn’t even feel like work. They’re just seeing pictures in their heads, and spending a couple of minutes a day looking at these pictures and asking themselves naïve questions.

对他们来说，这一切都很正常。他们显然感觉不到自己有什么天赋。

To them, it all looks completely normal. It certainly doesn’t feel like they have a gift.

12

没有花招

12

There Are No Tricks

这是 20 世纪 50 年代初美国的一个普通日子。一个普通家庭走在一条普通的道路上。父亲在开车，两个孩子坐在后座上。为了阻止他们争吵，父亲问了他们几个谜语：

It’s an ordinary day in the United States in the early 1950s. An ordinary family is on an ordinary road. The father is driving; the two kids are in the backseat. To stop them squabbling, the father asks them some puzzles:

1 + 2 + 3 + ... + 100 等于多少？

What is 1 + 2 + 3 + . . . + 100?

小儿子五岁。几秒钟后，他回答“5,000”。父亲告诉他差不多了。小男孩又思考了几秒钟，终于给出了正确答案：“5,050。”

The younger boy is five years old. In a few seconds he answers “5,000.” The father tells him that’s almost it. The young boy thinks for a few seconds more and finally gives the right answer: “5,050.”

这个五岁的男孩叫比尔·瑟斯顿。这个故事会让你开怀大笑，特别是如果你知道关于“数学王子”卡尔·弗里德里希·高斯（1777-1855）的著名故事的话。即使这个古老的故事只是一个传说，它也是众所周知的，瑟斯顿的父亲无疑听说过它。

The five-year-old boy is Bill Thurston. The story makes you smile, especially if you know the famous tale about Carl Friedrich Gauss (1777–1855), “the prince of mathematics.” Even if this old story is nothing but a legend, it’s very well known, and Thurston’s father undoubtedly had heard about it.

高斯是历史上最伟大的数学家之一，你可以毫不犹豫地将他与泰勒斯、毕达哥拉斯、欧几里得、阿基米德、花拉子米、笛卡尔、欧拉、牛顿、莱布尼茨、黎曼、康托尔、庞加莱、冯·诺依曼、格罗滕迪克和其他一些人相提并论。他才华横溢，富有创造力，以至于他的同时代人都拒绝相信他的智力来自一个生物学上正常的人类大脑。在某种程度上，他是那个时代的阿尔伯特·爱因斯坦。

Gauss was one of the greatest mathematicians in history, someone you could without hesitation place alongside Thales, Pythagoras, Euclid, Archimedes, al-Khwarizmi, Descartes, Euler, Newton, Leibniz, Riemann, Cantor, Poincaré, von Neumann, Grothendieck, and a few others. He was so spectacularly brilliant and creative that his contemporaries refused to believe that his intelligence came from a biologically normal human brain. He was in a way the Albert Einstein of his time.

事实上，它的结局与爱因斯坦的结局完全一样。高斯去世后，有人认为取走他的大脑是明智之举，希望能够揭开其中的秘密。两个世纪后，高斯的大脑仍然被珍藏在一个罐子里，保存在哥廷根大学的某个地方。至今还没有人发现任何特别有趣的东西。

And, in fact, it ended exactly as with Einstein. When Gauss died, someone thought it wise to take his brain in the hope of uncovering its secrets. Two centuries later, Gauss’s brain is still preciously conserved in a jar, somewhere in the collections at the University of Göttingen. No one has of yet found anything particularly interesting to say about it.

传说，七岁那年，年轻的高斯吓坏了他的老师。后者要求全班计算从 1 到 100 的整数之和，以为这样他就能有整整十五分钟的安宁时间。他没想到其中一个孩子会在几秒钟内找到答案。

The legend has it that, at the prime age of seven, the young Gauss scared the hell out of his instructor. The latter had asked the class to calculate the sum of whole numbers from 1 to 100, in the belief that he’d be giving himself a good quarter of an hour’s peace. He hadn’t counted on one of the kids finding the answer in seconds.

我十七岁的时候，高中数学老师给我们讲了这个故事，给我们留下了很深的印象。我们当时想不通高斯怎么能算得这么快。面对这样的天才，我们都觉得自己很可怜。

I was seventeen when our senior high school math teacher told us this story, which had a big impression on us. We couldn’t figure out how Gauss was able to calculate so quickly. Faced with such genius, we all felt ourselves rather pathetic.

老师给我们的解释是，有一个“技巧”。你想计算从 1 到 100 的整数，也就是说，把它们加起来：

The explanation that our teacher gave us was that there was a “trick.” You want to calculate the whole numbers from 1 to 100—that is, to add them up:

1 + 2 + 3 + 4 + . 。 。 + 97 + 98 + 99 + 100

1 + 2 + 3 + 4 + . . . + 97 + 98 + 99 + 100

这个魔术是将每个整数数两次，然后将两个总数放在两条线上，使总数翻倍，方法如下：

The trick consists of doubling this sum by counting each whole number twice and placing the two sums on two lines, in the following manner:

       1 + 2 + 3 + 4 + . 。 。 + 97 + 98 + 99 + 100

       1 +   2 +   3 +   4 + . . . + 97 + 98 + 99 + 100

+ 100 + 99 + 98 + 97 + . 。 。 + 4 + 3 + 2 + 1

+ 100 + 99 + 98 + 97 + . . . +   4 +   3 +   2 +     1

多么奇怪的想法！为什么要把每个数字放两次？为什么要以这种奇怪的方式将它们一个接一个地放在另一个上面？也许这很奇怪，但你有权这样做。无论如何，每个数字从 1 到 100 确实出现了两次。因此，大数额的值是我们要找的数字的两倍。

What a strange idea! Why put down each number twice? Why place them one above the other in this bizarre fashion? Maybe it is strange, but you have the right to do it. At any rate, each number from 1 to 100 does appear twice. The value of the big sum is thus double the number that we’re looking for.

现在，我们不要看行，而是看列。一共有 100 列，每列有两个数字，它们的和始终是 101。这看起来像魔术，但这是真的。因此，最大的和等于 100 × 101，即 10,100。我们要找的数字是它的一半，即 5,050。

Now instead of looking at the lines, look at the columns. There are 100 columns and in each there are two numbers whose sum is always 101. It seems like magic, but it’s true. The big sum thus equals 100 × 101, or 10,100. The number we’re looking for is half of that, or 5,050.

不要因为必须反复阅读这个推理才能相信它而感到羞愧。与所有数学推理一样，它有些奇怪和令人生畏。一开始你必须逐行解读它，这需要花费大量的时间和精力。

Don’t be ashamed of having to read over this reasoning a number of times before you find it convincing. As with all mathematical reasoning, there’s something bizarre and intimidating about it. At first you have to decipher it line by line, which takes a lot of time and effort.

然而，推理的步骤相当简单，应该能让你得出以下三个结论：

The steps of the reasoning, however, are simple enough, and should allow you to reach these three conclusions:

1. 这是“1 到 100 的整数之和等于 5,050”这一事实的有效证明。

1. It’s a valid proof of the fact that the sum of whole numbers from 1 to 100 equals 5,050.

2. 可以相信，心算能力强的人只需几秒钟就能在脑子里做出这一推理。

2. It’s believable that someone quick in mental calculations could do this reasoning in their head in a few seconds.

3. 但如此疯狂的想法怎么会出现在七岁小孩高斯的脑袋里呢？

3. But how on Earth could such a crazy idea arise in the head of the seven-year-old Gauss?

无论如何，这些都是我 17 岁时得出的结论。我发现数学不适合我，因为它是为那些其他人、那些天才准备的，他们的大脑与我不同，可以想出如此不可思议的想法。

At any rate, those were the conclusions I made myself when I was seventeen. I figured out that math wasn’t for me because it was meant for those other people, those geniuses, whose brains worked differently than mine and could come up with such incredible ideas.

我的老师是一位出色的教练，我很感激他教给我的一切。但那天，他告诉我们有一个“窍门”，这传达了错误的信息。

My teacher was an excellent instructor, and I’m grateful for everything he taught me. But that day, by telling us there was a “trick,” he sent us the wrong message.

不存在任何诡计。从来没有，也永远不会有。相信诡计的存在与相信本质上违反直觉的真理的存在一样有害。这是系统 2 教条的两个核心迷信，这种信念认为我们的学费不值一提，我们必须机械地应用我们不完全理解的方法。

There are no tricks. There never were any and there never will be. Believing in the existence of tricks is as toxic as believing in the existence of truths that are counterintuitive by nature. These are the two central superstitions of the System 2 dogma, this belief that our intuition isn’t worth a dime and that we have to mechanically apply methods that we don’t fully understand.

当然，有些事情我们可能不明白为什么会发生。这种情况经常发生。但这总是暂时的情况，只是在等待一个解释。

Of course it can happen that things work without our understanding why. It happens often enough. But it’s always a temporary situation that’s just waiting for an explanation.

相信技巧的存在，就等于接受了这样一种观点：有些东西你永远无法理解，你必须死记硬背。这等于混淆了逐行验证证明和直观理解证明。这等于对系统 2 产生一种服从关系。这等于接受一种对你来说非常不公平和羞辱的角色划分：伟大的天才会发现技巧，而你只擅长检查所有东西是否合情合理。

Believing that tricks exist is to accept the idea that there are things you’ll never understand and that you have to learn by heart. It’s to confuse the line-by-line verification of a proof with its intuitive understanding. It’s to enter into a submissive relationship to System 2. It’s to accept a division of roles that is deeply unfair and humiliating for you: the great geniuses find the tricks, while you’re only good for checking that it all adds up.

坦白说，我根本不关心验证 1 到 100 的整数之和是否真的等于 5,050。我想知道的，也是我们所有人都想知道的，是如何像高斯和瑟斯顿那样思考。

Frankly, I couldn’t care less about verifying that the sum of whole numbers from 1 to 100 really equals 5,050. What I want to know, what we all want to know, is how to think like Gauss and Thurston.

语言陷阱

The Language Trap

要了解这些数学“技巧”背后隐藏着什么，最简单的方法就是遵循香蕉面包的食谱。

To understand what’s hidden behind these math “tricks,” the simplest thing is to follow a recipe for banana bread.

原料：

Ingredients:

1 1/2 杯（195 克）通用面粉

1 1/2 cups (195 grams) all-purpose flour

1茶匙小苏打

1 teaspoon baking soda

1/4 茶匙细海盐

1/4 teaspoon fine sea salt

3/4 茶匙肉桂粉

3/4 teaspoon ground cinnamon

3 根中号香蕉

3 medium bananas

8 汤匙（115 克或 1 条）无盐黄油，融化并冷却

8 tablespoons (115 grams or 1 stick) unsalted butter, melted and cooled

3/4杯（150克）浅红糖

3/4 cup (150 grams) packed light brown sugar

2 个大鸡蛋，轻轻打散

2 large eggs, lightly beaten

1茶匙香草精

1 teaspoon vanilla extract

使用方法：

Directions:

1. 在一个大碗里，用叉子将香蕉捣碎。

1. In a large bowl, mash the bananas with a fork.

2. 将面粉、小苏打和盐混合。

2. Combine the flour, baking soda, and salt.

3. 将鸡蛋和糖搅拌在一起。

3. Cream together the eggs and sugar.

4. 加入捣碎的香蕉、香草、黄油和肉桂，搅拌均匀。

4. Stir in the mashed bananas, vanilla, butter, and cinnamon.

5. 加入面粉混合物，每次三分之一，搅拌直至混合均匀。

5. Stir in the flour mixture, a third at a time, until just combined.

6. 将面糊倒入一个 9×5 英寸的面包盘中。在 350 华氏度的温度下烘烤约 1 小时。

6. Pour the batter into a 9-by-5-inch loaf pan. Bake for about 1 hour at 350 degrees F.

想象一下这个菜谱的不同步骤：

Visualize the different steps of this recipe:

—你先买香蕉。香蕉在你手里。你去收银台付款。你能想象出香蕉的模样吗？

—You start by buying the bananas. They’re in your hand. You go to the register to pay for them. Can you picture them?

—您正处于第 1 步。香蕉在碗里。您手拿叉子，准备将香蕉捣碎。您还能想象出它们的模样吗？

—You’re at step 1. The bananas are in the bowl. You have a fork in your hand and you’re getting ready to mash them. Can you still picture them?

在这两个步骤之间，你的心理形象已经转换到了另一个层面。在捣碎香蕉之前，你已经在心理上剥了香蕉皮。所谓的“花招”背后，通常都有这样的操作：瞬间改变心理形象，出于“显而易见”的原因，但其他人可能并不那么明显。

Between these two steps, you’ve switched to a different mental image. Just before mashing the bananas, you’ve mentally peeled them. Behind the so-called “tricks,” there’s generally an operation of this kind: a change of mental image that’s done in a flash, for “obvious” reasons that might not be so obvious to anyone else.

如果你熟悉香蕉，那么在捣碎之前必须先剥皮。但如果你以前从未见过香蕉，就没那么明显了。食谱从不记录你需要采取的所有步骤。总会有一些细节被遗漏，也就是著名的“技巧”。这就是为什么这么多人更喜欢看烹饪视频而不是阅读食谱。

When you’re familiar with bananas, it’s obvious that you have to peel them before you mash them. But if you’ve never seen a banana before, it’s not so obvious. Recipes never capture all the steps you need to take. There are always some missing details, the famous “tricks.” It’s why so many people prefer watching a cooking video to reading a recipe.

你从小就熟悉香蕉。你甚至可以说，你与香蕉建立了某种精神上的亲密关系。你知道很多关于它们的事情，但从未告诉过任何人。即使你不知道它的名字，但你能背出沿着果肉延伸的绳子。你从来没有用这根绳子做过任何事情，但它的外观和特性总是让你印象深刻。你也知道，尽管你从未敢说出来，但地球上没有任何东西能像香蕉肉一样如此柔软和令人满意地被挤压。香蕉这个词不仅会唤起一个心理形象，而是会唤起多种可能的心理形象。瞬间，甚至无需尝试，也无需任何人告诉你如何去做，你总能选择正确的形象。不剥皮就捣碎香蕉是如此愚蠢，你觉得它很有趣。这种蠢事只有机器人才会做。

You’ve been acquainted with bananas since childhood. You could even say that you’ve developed some kind of spiritual intimacy with them. You know a lot of things about them that you’ve never told anyone. You know by heart the string that runs along the flesh even if you don’t know its name. You’ve never done anything with this string, but it’s always struck you by its appearance and properties. You also know, without ever having dared say it, that nothing on Earth squishes in such a soft and satisfying manner as the flesh of a banana. The word banana doesn’t evoke just one mental image but a multitude of possible mental images. Instantly, without even trying, and without anyone telling you how to do it, you always pick the right image. Mashing bananas without peeling them is so idiotic that you find it funny. It’s the stupid kind of thing that only a robot would do.

当高斯或瑟斯顿想要把 1 到 100 的整数相加时，他们会选择正确的方法来形象化这些数字，也就是让计算变得更容易的方法。他们立刻就找到了方法，毫不费力，也没有人告诉他们怎么做。他们知道如何调动对数字的熟悉，就像你知道如何调动对香蕉的熟悉一样。这是完全相同的智慧。

When Gauss or Thurston wanted to add up the whole numbers from 1 to 100, they picked the right way to visualize these numbers, the way that made the calculation easier. They found it instantly, without any effort, and with no one to tell them how. They knew how to mobilize their familiarity with numbers in the same way that you know how to mobilize your familiarity with bananas. It’s exactly the same kind of intelligence.

在数学中，突然发生的奇迹或似乎凭空而来的想法总是表明你缺少一个形象。你看待事物的方式不正确。缺少了一些东西。存在一种更好的方式，更简单、更清晰、更深入，你还不知道，也许没有人知道。寻找​​和发现看待事物的正确方式是数学的驱动力。这是你从中获得乐趣的主要来源。

In mathematics, the sudden occurrence of a miracle or an idea that seems to come out of nowhere is always the signal that you’re missing an image. Your way of looking at things isn’t the right one. Something is missing. There exists a better way, simpler, clearer, deeper, that you don’t know yet and that, perhaps, no one yet knows. Looking for and finding the right way of seeing things is the driving force of mathematics. It’s the main source of pleasure you can take from it.

每次有人跟你谈论“技巧”时，他们都会告诉你，在事情开始变得有趣时，要停止思考。

Each time someone talks to you about “tricks,” they’re telling you to stop thinking at precisely the moment when it starts to get interesting.

讽刺的是，在你刚刚熟悉香蕉的同时，在你童年的那个遥远的时光里，你也开始熟悉数字。如果你没有发展出正确的水平如果你对数字没有那么熟悉，你永远也不会学会如何计数。

The irony of all this is that while you were just getting familiar with bananas, at that faraway time of your childhood, you were also getting familiar with numbers. If you hadn’t developed the right level of intimacy with numbers, you would never have been able to learn how to count.

不幸的是，你从此就失去了与数字的这种亲密关系。童年之后，你陷入了我所说的语言陷阱，这使你无法像高斯或瑟斯顿那样“看到”从 1 到 100 的整数之和。

Unfortunately, you’ve since lost this intimate relationship to numbers. Following your early childhood, you fell into what I call the language trap, which is what stops you from “seeing” the sum of whole numbers from 1 to 100 like Gauss or Thurston.

语言陷阱是指人们相信只要命名事物就足以使它们存在，我们无需费力去真正想象它们。

The language trap is the belief that naming things is enough to make them exist, and we can dispense with the effort of really imagining them.

这种信念是系统 2 意识形态的典型代表。我们被告知，我们应该用语言思考，渴望超越语言是一种白日梦。这种捷径是有问题的，甚至完全是谎言。命名事物当然可以让我们唤起它们，但不能让它们以允许创造性思维的强度和清晰度出现在我们的脑海中。

This belief is typical of the ideology of System 2. We’re told that we should think with words and that yearning to move beyond words is a pipe dream. This shortcut is problematic, if not an outright lie. Naming things certainly allows us to evoke them, but not to make them present in our mind with the intensity and clarity that allow for creative thinking.

“不要去想粉红色的大象。”这被认为是一个语言悖论，因为这句话本身迫使我们去想粉红色的大象。只是这种被动、不情愿的思考粉红色大象的方式并不能让你去了解和真正理解它们。试着想象一头真人大小的粉红色大象站在你面前。花点时间看看它，仔细研究它。这个有意为之的形象将比你在本段开头脑海中形成的模糊形象更加深刻、更加引人入胜、更加精确。当你自由发挥你的想象力时，它几乎是没有限制的。

“Don’t think of a pink elephant.” This is considered a linguistical paradox, since the sentence itself forces us to think of a pink elephant. Except that this passive, reluctant way of thinking about pink elephants isn’t one that will allow you to get to know and really understand them. Try to imagine a life-sized pink elephant standing before you. Take the time to look at it and study it closely. This intentional image will be incredibly more profound, more absorbing, more precise than the fuzzy image formed in your mind at the beginning of this paragraph. When you give free rein to your imagination, it is nearly without limits.

正是这种想象力让你摆脱语言陷阱，解决数学问题。这项活动是系统 3 的核心。它意味着要刻意地去观察，毫无保留，毫不妥协，全身心投入。

It’s this effort of the imagination that allows you to get out of the language trap and solve mathematical problems. This activity is at the heart of System 3. It implies deliberately trying to see, without reserve or half measures, with a full physical commitment.

当您读到“1 到 100 之间的整数之和”时，如果您满足于脑海中形成的模糊图像，那么您实际上什么也看不到。

When you read “the sum of whole numbers from 1 to 100,” if you content yourself with the fuzzy image that forms in your head, you won’t really see anything.

不要让自己被文字所迷惑，要强迫自己认为总数就在你面前。强迫自己想象从 1 到 100 的整数以物理形式呈现，在现实世界中显现，精心排列在你面前。如果你设法看到它们，并花时间仔细检查场景，你就会找到一种方法来计算它们的总和。

Instead of letting yourself be lulled by words, force yourself to think that the sum is physically present in front of you. Force yourself to imagine the whole numbers from 1 to 100 in physical form, made manifest in the real world, carefully lined up in front of you. If you manage to see them and you take the time to carefully examine the scene, you’ll find a way to calculate their sum.

为了让您有机会自己找到它，我建议您在继续之前稍事休息。

To give you a chance to find it for yourself, I recommend you take a short break before continuing.

总体情况

The Big Picture

在第 6 章引用的文本“关于数学的证明和进展”中，瑟斯顿给出了一些关于数学对象大小的令人惊讶的建议——我从未在其他任何地方读到过。

In “On Proof and Progress in Mathematics,” the text cited in chapter 6, Thurston gives some surprising advice—that I’ve never read anywhere else—about the size of mathematical objects.

当我们在脑海中想象它们时，我们可以选择将它们视为“手中的小物体”，或“更大的人体结构”，或“包围我们并在其中移动的空间结构”。从逻辑的角度来看，这应该没有任何区别。然而，瑟斯顿说，大小非常重要：“我们倾向于在更大的范围内更有效地思考空间意象：就好像我们的大脑更认真地对待更大的事物，并能为它们投入更多的资源。”

When we imagine them in our heads, we can choose to see them as “little objects in our hands,” or as “bigger human-sized structures,” or as “spatial structures that encompass us and that we move around in.” From a logical standpoint, it shouldn’t make any difference. Thurston, however, says size is quite important: “We tend to think more effectively with spatial imagery on a larger scale: it’s as if our brains take larger things more seriously and can devote more resources to them.”

如果人们坚信自己没有任何几何直觉，只是错误地想象出太小的图形，以致于无法看到任何东西，那该怎么办？

What if people who are convinced that they don’t have any geometric intuition simply make the mistake of imagining figures that are too small, so that it’s impossible to see anything?

无论如何，瑟斯顿的评论非常适用于大象：想象一只你可以握在手中的小象，现在想象一只真人大小的大象，它看起来不太开心，你也不想引起它的注意：它会以完全不同的方式调动你的认知资源。

At any rate, Thurston’s remark applies very well to elephants: imagine a tiny elephant that you can hold in your hand, and now imagine a life-sized elephant, who doesn’t look happy, and whose attention you don’t want to attract: it mobilizes your cognitive resources in an entirely different manner.

语言陷阱是这一现象的极端版本瑟斯顿 (Thurston) 描述了这一概念。像“1 到 100 之间的整数之和”这样的表达式是一种表示非常精确的数学对象的方便方法。它允许你谈论它，但它也是一种摆脱它的方法，将它放在一定距离，这样它就不会再困扰你了。

The language trap is the extreme version of the phenomenon described by Thurston. An expression like “the sum of whole numbers from 1 to 100” is a convenient way to designate a very precise mathematical object. It allows you to speak of it, but it’s also a way of getting rid of it, to put it at some distance, so that it doesn’t bother you any longer.

你以为你看到了总数，但实际上你并没有。你感觉不到它的存在。你没有认真对待它。

You think you see the sum, but you don’t really. You can’t feel its looming presence. You don’t take it seriously.

这个总数也可以写成 5,050。十进制书写的一大优势是简洁。它独立、实用、易说易写。这种心理表示法的优点中也有缺点：数字被放在如此远的地方，以至于变得微不足道，几乎看不见。

This sum can also be written as 5,050. The big advantage of decimal writing is that it’s compact. It’s discrete, practical, easy to say and easy to write. This mental representation has its weakness in its strength: the number is put at such a distance that it becomes minuscule, almost invisible.

数学方程式总是表明，两个表面上不同的文字实际上代表同一个物体。如果你让自己沉迷于语言，如果你将单词与它们所代表的物体混淆，你就没有机会“看到”数学方程式。

A mathematical equation always states that two writings that are different in appearance designate in reality one and the same object. If you let yourself be lulled to sleep by language, if you confuse words with the objects they designate, you don’t give yourself any chance to “see” mathematical equations.

实现这一目标的唯一方法就是超越文字。将“1 到 100 的整数之和”替换为“1 + 2 + 3 + . . + 98 + 99 + 100”是一个好的开始。您可能会觉得总和更加直观和具体。但那只是一种幻觉。实际上，您会错过大多数数字，即那些被省略号隐藏的数字。数学符号就像文字一样，它们属于语言。您还需要超越它们。

The only way to get there is to go beyond words. Replacing “the sum of whole numbers from 1 to 100” with “1 + 2 + 3 + . . . + 98 + 99 + 100” is a good start. You might have the impression of seeing the sum in a more tangible and concrete way. But that’s only ever an illusion. In reality, you’ll be missing most of the numbers, those hidden by the ellipses. Mathematical symbols are like words, they belong to language. You also need to get beyond them.

为了不走捷径或缩写，全面地看待它，认真对待它并给予它应有的地位，你需要想象它就存在于你的面前，具有真人大小。

To see the sum in its entirety, without shortcuts or abbreviations, to take it seriously and give it the place it deserves, you need to imagine it as physically present, life-sized, right in front of you.

在想象总和之前，让我们先从一个数字开始，例如 3。在物理世界中想象数字 3 很容易：你只需要想象三个物体，就像在小学时老师要求学生在脑海中想象三个橙子一样。 这种与数字的童真关系，这种与抽象事物进行身体互动的需要，是做数学题的正确心态。通过看到三个橙子代替数字 3，你开始摆脱语言陷阱。你不再将数字的书写与它的值混淆。

Before imagining the sum, let’s start with a single number, for example, 3. Imagining the number 3 in the physical world is easy enough: you just have to imagine three objects, like in primary school when the teacher asks students to picture three oranges in their head. This childlike relationship with numbers, this need for a bodily interaction with abstract things, is the right state of mind to do mathematics. By seeing three oranges in place of the number 3, you begin freeing yourself from the language trap. You stop confusing the writing of a number with its value.

数学家有句俗语：整数总是有意义的。然而，要想象从 1 到 100 的整数之和，我建议不要使用橙子：你会发现自己有很多橙子——它们会洒得到处都是，一片狼藉。

Mathematicians have a saying that a whole number always counts for something. However, to imagine the sum of whole numbers from 1 to 100, I’d advise not using oranges: you’d find yourself with a lot of oranges—they will spill all over the place and it will be a mess.

就我个人而言，我觉得用立方体来想象这个场景更容易。我能够将每个数字想象成一堆立方体，并将这些立方体并排排列，从 1 到 100。

Personally, I find it much easier to imagine the scene using cubes. I’m able to visualize each number as a pile of cubes and to arrange these piles side by side, from 1 to 100.

很难准确地画出我脑海中所看到的东西。我的脑海中的图像并不完全清晰，而且这堆东西太大了，无法放在纸上。因此，我只能画一个近似值。从正面看，它看起来应该是这样的：

It’s hard to draw exactly what I see in my head. My mental image isn’t entirely distinct, and the pile would be too big to put on the page. I’m only able, therefore, to draw an approximation. Seen head on, it would look something like this:

画错了，但那不重要。重要的是知道错在哪里。在这种情况下，缺少了一些立方体。这堆立方体应该有 100 个立方体，而不是像我的画中那样有 18 个立方体横跨和 18 个立方体高和一百个立方体的高度。你需要记住这一点。尽管如此，这幅画对我来说似乎是一种分享我心理形象的好方法（如果我画出所有的立方体，它们就太小了）。

The drawing is wrong, but that doesn’t matter. What matters is to know in what way it is wrong. In this case, there are some cubes missing. Instead of having eighteen cubes across and eighteen cubes high, as in my drawing, the pile should have one hundred cubes across and one hundred cubes high. You need to keep that in mind. Despite that, this drawing seems to me to be a good way to share my mental image (if I drew all the cubes, they would be too tiny).

瞧！我们完成了。是不是很简单？

Voilà! We’re done. Wasn’t that easy?

数学家倾向于认为，一旦他们觉得头脑中形成了正确的图像，证明就完成了，就像国际象棋选手在看到一名选手处于获胜位置时在将死之前停止游戏一样。

Mathematicians tend to think that a proof is finished once they feel that the right image is formed in their heads, like when chess players stop the game before checkmate once they see one player has a winning position.

但是让我们花点时间来完成这场比赛，因为即将到来的将死对你来说可能还不那么明显。

But let’s take the time to finish the game, as the coming checkmate may not yet be that obvious to you.

如果你脑子里有这个图像，就很难不看到三角形。你要找的数字，也就是立方体的总数，就是三角形的面积。有一个简单的小学公式来计算面积。这里有两种方法来完成游戏，取决于你是否知道这个公式。

If you have this image in your head, it’s difficult not to see a triangle. The number you’re looking for, that is, the total number of cubes, is the area of the triangle. There is a simple primary school equation to calculate the area. Here are two ways to finish the game, depending on whether you know the equation.

1.你知道这个公式。要计算三角形的面积，你需要将底边乘以高，然后除以 2。这里底边是 100，高也是 100。将它们相乘得到 10,000，再除以 2 得到 5,000。

1. You know the equation. To calculate the area of a triangle, you multiply the base by the height and divide by 2. Here the base is 100 and the height is 100. Multiplying them gives us 10,000 and dividing that by 2 gives us 5,000.

我们快到了。我们只是犯了瑟斯顿五岁时犯的同样错误。这是个好兆头；我们肯定走在正确的道路上。

We’re almost there. We just made the same error Thurston did when he was five. That’s a good sign; we’re surely on the right track.

错误在于我们忘记了对角线上方的半立方体，它们不计入三角形的面积。我们忘记了 100 个半立方体，所以我们必须加上 50，这样就是 5,050。

The error was forgetting about the half cubes that are above the diagonal and aren’t counted in the area of the triangle. We’ve forgotten 100 half cubes, so we have to add 50, which makes 5,050.

2.你不知道这个等式。别担心，你会重新发明它。仔细观察，你会发现三角形是矩形的一半。如果你拿出最初的三角形堆（白色）和这个三角形堆的副本（灰色），然后将副本翻转并放在最初的堆顶部，你会得到如下结果：

2. You don’t know the equation. No worries, you’re going to reinvent it. By looking carefully, you can see that a triangle is a half of a rectangle. If you take your initial triangular pile (in white) and a copy of this triangular pile (in gray), and then turn the copy and put it on top of the initial pile, you come up with something like this:

这样，你就得到了一个长方形，宽 100 个立方体，高 101 个立方体，因此由 100 x 101 = 10,100 个立方体组成。因此每个三角形中有 5,050 个立方体。著名的“技巧”就是将两个总和相加，这只不过是一种将矩形面积分解为两个三角形的方法：

You thus get a rectangle 100 cubes across and 101 cubes high, therefore formed of 100 x 101 = 10,100 cubes. There are thus 5,050 cubes in each triangle. The famous “trick” that consists of taking the two sums and putting them one over the other is nothing more than that, a way of breaking down the area of the rectangle into the two triangles:

       1 + 2 + 3 + 4 + . 。 。 + 97 + 98 + 99 + 100

       1 +   2 +   3 +   4 + . . . + 97 + 98 + 99 + 100

+ 100 + 99 + 98 + 97 + . 。 。 + 4 + 3 + 2 + 1

+ 100 + 99 + 98 + 97 + . . . +   4 +   3 +   2 +     1

概率功夫

Probabilistic Kung Fu

球和球棒，1 到 100 的整数之和：我喜欢这些问题，因为它们很简单，但它们却完全说明官方数学（语言的囚徒）和秘密数学（你头脑中所做的事情）之间的差距。

The ball and the bat, the sum of whole numbers from 1 to 100: I love these problems because they are elementary, and yet they fully illustrate the gap that separates official mathematics, the prisoner of language, and secret mathematics, what you do inside your head.

在这两种情况下，简单的可视化努力就足以让 99％ 的人觉得根本不容易的事情变得简单。

In both cases, a simple visualization effort suffices to make something easy that 99 percent of people don’t find easy at all.

事情并不总是那么简单。光有形象化并不总是足够的，问题不在于剥夺你机械的演绎推理。为了理解数学，你必须训练自己将想象力和语言、直觉和逻辑、沉思和计算结合起来，既能看到大局，也能看到细节。

It’s not always so simple. Visualization isn’t always enough, and the issue isn’t to deprive yourself of mechanical deductive reasoning. In order to understand mathematics, you have to train yourself to bring together imagination and language, intuition and logic, reverie and calculation, seeing both the big picture and the details.

我也不想给人留下这样的印象：数学问题总是数字的，所有的直觉本质上都是几何的。

Nor do I want to give the impression that math problems are always numerical and that all intuition is geometric in nature.

数学对象本质上非常多样化，直观理解数学对象需要调动不同的思维资源。该表列举了几个主要的数学领域。虽然不完整且过于简单，但它可以让你大致了解数学。

Mathematical objects are very diverse in nature, and their intuitive comprehension mobilizes different mental resources. The table enumerates several of the main mathematical domains. It’s incomplete and oversimplified, but it gives you a general idea.

这些领域都有各自的词汇和直觉。它们就像是对应着我们身体的不同使用方式、不同的我们大脑中的不同区域，以及我们集中注意力的不同方式。他们可能给人的印象是他们在谈论不同的事情，但事实上他们只是对同一个现实，即数学现实，提出了不同的观点。当你体验到它时，数学的统一性绝对是令人着迷的。它甚至可能令人不知所措。

These domains each have their own vocabulary and intuitions. It’s like they correspond to different ways of using our bodies, different regions in our brains, different ways of focusing our attention. They might give the impression that they’re talking about different things, but in fact they’re simply bringing different points of view to the same reality, the mathematical reality. When you get to experience it, the unity of mathematics is absolutely fascinating. It can even be overwhelming.

数学的主要领域

The main areas of mathematics

很多时候，数学发现仅仅是两种不同类型的直觉之间的桥梁。

Very often, mathematical discoveries are merely bridges between two intuitions of different kinds.

从最基本的层面上讲，这就是我们上面刚刚做的事情：一个几何公式（给出三角形或矩形的面积）使我们能够解决算术问题（从 1 到 100 的整数之和）。

At a very elementary level, this is what we’ve just done above: a geometry formula (that gives the area of a triangle or rectangle) allowed us to solve an arithmetic problem (the sum of whole numbers from 1 to 100).

让我们用另一个更加引人注目的例子来结束本章。

Let’s conclude this chapter with another, even more striking example.

如果您无法想象眼前 1 到 100 的整数，那么有一种更简单的方法。与其费力地处理 1 到 100 的所有数字，为什么不随机选取其中一个呢？当您随机选取 1 到 100 之间的数字时，它的平均值是多少？

If you have problems visualizing whole numbers from 1 to 100 life-sized before you, there’s an easier way. Instead of tiring yourself out by dealing with all the numbers from 1 to 100, why not take only one of them, at random? When you randomly choose a number between 1 and 100, what is its value on average?

如果你觉得这太抽象了，那么可以想象一下。你正在参加一个游戏节目。一个袋子里有一百张支票：一张 1 美元的支票，一张 2 美元的支票，等等，直到一张 100 美元的支票。闭上眼睛，你只能选一张支票。平均而言，你预计能赢多少钱？

If that seems abstract to you, here’s a concrete way of imagining it. You’re on a game show. In a bag, there are one hundred checks: a check for $1, a check for $2, and so on up to a check for $100. You can pick only one check, eyes closed. On average, how much do you expect to win?

我重复一下这个问题：当你从 1 到 100 中随机选择一个数字时，它的平均值是多少？

I’ll repeat the question: when you choose a number at random from 1 to 100, what is its value on average?

大多数人会不假思索地回答“50”。对他们来说，这似乎很明显。但如果从 1 到 100 的整数平均值是 50，那么它们的总和一定是 5,000：100 个数字的总和是它们平均值的 100 倍。这对大多数人来说也是显而易见的。

Most people will answer “50” without thinking. It just seems obvious to them. But if the average of whole numbers from 1 to 100 is 50, then their sum must be 5,000: the sum of 100 numbers is 100 times their average. That’s also obvious to most people.

那么，是什么让人们不假思索地回答呢？就像五岁的瑟斯顿那样，1 到 100 的整数之和是 5,000？

So what is it that keeps people from answering without thinking, like the five-year-old Thurston did, that the sum of whole numbers from 1 to 100 is 5,000?

（如果你的直觉告诉你平均值是 50，那它就有点不对了。别担心，这正是瑟斯顿所犯的错误。平均值实际上是 50.5。在这个阶段，1% 的错误不应该破坏你的乐趣。）

(If your intuition does tell you that the average is 50, it’s slightly wrong. Don’t worry, it’s exactly the same mistake Thurston made. The average is actually 50.5. At this stage, an error of 1 percent shouldn’t spoil your fun.)

刚刚发生了什么？这怎么可能？问题的难度怎么会突然消失，就像蒸发了一样？如果你觉得这很荒谬，那是因为你低估了概率思维的力量。在脑子里组装 5,050 个立方体是一项艰巨的任务，几乎感觉你需要一辆叉车。相比之下，概率方法是一种功夫，它让你的注意力集中在一个数字上，而你的潜意识会完成所有繁重的工作。

What just happened? How is it even possible? How come the difficulty of the problem vanished all of a sudden, like it was vaporized? If this seems absurd to you, it’s because you’re underestimating the power of probabilistic thinking. Assembling 5,050 cubes in your head is a demanding task, it almost feels like you need a forklift. In contrast, the probabilistic approach is a sort of kung fu that focuses your attention on a single number while your subconscious does all the heavy lifting.

事实上，即使你没有意识到，你已经知道如何计算 1 到 100 之间的整数总和。

In fact, you already knew how to calculate the sum of whole numbers from 1 to 100, even if you were not aware that you did.

平均值的概念纯粹是人类的发明，是一个抽象的数学概念，它是你被教导的，并且你已经吸收到了你内心最深处，就像十进制书写一样。你已经学会了“看到”平均值，也就是说，无需思考或写下来就可以计算它们。如果你想验证你的直觉并将其转化为严谨的思考，如果你真的想理解为什么平均值是 50.5 而不是 50，你需要倾听你自己，倾听你的潜意识过程及其机制。

The notion of average is a purely human invention, an abstract mathematical concept that you were taught and that you’ve assimilated in the deepest part of yourself, just like decimal writing. You’ve learned to “see” averages, that is, to calculate them without even thinking or having to write them down. If you want to validate your intuition and transform it into rigorous thinking, if you really want to understand why the average is 50.5 rather than 50, you need to listen to yourself, to your unconscious processes and their mechanisms.

这种内省工作是数学的核心。它意味着解构你不假思索使用的心理图像，并确定你可以改进的地方。如果做得正确，这种做法将日复一日地强化你的直觉。

This introspective work is at the heart of mathematics. It implies deconstructing the mental images that you use without thinking and identifying where you can improve them. Done properly, this practice will reinforce your intuition day after day.

数学家操纵抽象概念，而他们忘记了抽象概念的抽象本质，他们更喜欢称其为对象。他们还喜欢说这些对象是存在的。这样说，他们并不一定想参与自柏拉图以来就抽象概念的现实性展开的古老形而上学辩论。他们只是想说这就是他们做数学的方式：通过创建与这些对象的熟悉联系，让自己在头脑中想象和操纵它们，就像你操纵香蕉一样。

Mathematicians manipulate abstractions whose abstract nature they’ve forgotten, and that they prefer to call objects. They also like to say that these objects exist. By saying that, they don’t necessarily want to take part in the old metaphysical debate that since Plato has argued over the reality of abstractions. They simply want to say that that’s how they do math: by creating familiar ties to these objects, letting themselves imagine and manipulate them in their heads exactly as you would a banana.

要真正了解一个数学对象，你必须长时间观察它，保持专注和超然，保持好奇心和开放心态。你需要花时间玩它，建立一种亲密的关系，一种超越语言的关系。

To get to really know a mathematical object, you have to observe it for a long time, with intensity and detachment, with curiosity and open-mindedness. You need to take the time to play with it and create an intimate relationship, a relationship that takes place outside of language.

当爱因斯坦说他“充满好奇心”时，当格罗滕迪克说他“独自一人聆听各种事物，全神贯注于孩子的游戏”时，他们谈论的就是这个。

When Einstein said he was “passionately curious,” when Grothendieck said he was “alone and listening to things, intensely absorbed in a child’s game,” this is what they were talking about.

13

看起来像个傻瓜

13

Looking Like a Fool

当我刚开始读本科时，我认为数学创造力是比我聪明的人的专利。我认为数学智力是天生的，每个人都有预先确定的数量。我很幸运，能比平均水平高一点。天才们得到的数学智力要高得多。

When I started as an undergrad, I thought that mathematical creativity was reserved for people smarter than me. I thought that mathematical intelligence was innate and that everyone received a predetermined quantity. I was lucky enough to get a bit above the average. The geniuses got scandalously more.

我当时还不明白数学智力是你自己构建的。它是每个人都可以自由练习的一种体力活动的自然副产品：数学想象力。

I hadn’t yet understood that mathematical intelligence was something you constructed for yourself. It’s the natural byproduct of a physical activity that everyone is free to practice: mathematical imagination.

数学是一门想象力的科学。在那些允许自己想象、观察和操纵数学对象的人和那些不这样做的人之间存在着巨大的鸿沟。多年来，这种鸿沟变得可怕、可憎，几乎就像拥有满屋子玩具和游戏的孩子和甚至不知道玩具存在的孩子之间的鸿沟一样可怕和可憎​​。

Mathematics is the science of imagination. Between those who allow themselves to imagine, observe, and manipulate mathematical objects and those who don’t, there’s an enormous divide. Over the years, this divide becomes monstrous, obscene, almost as monstrous and obscene as the divide between children with a room full of toys and games, and those who don’t even know toys exist.

与普遍的看法相反，逻辑并不是想象力的敌人。它甚至可以成为亲密的盟友。想象力的真正敌人是恐惧，它阻碍了理解，让我们觉得自己像个傻瓜。

Contrary to popular belief, logic isn’t the enemy of imagination. It can even be a close ally. The real enemy of imagination, that which blocks understanding and makes us feel like fools, is fear.

恐惧是我们真正的局限。它与我们所有人有关，无论在哪个层次，从最差的到最好的，从初学者到著名的学者。我们都有盲点，这些词的简单说出让我们感到恐惧，因为我们把它们与我们最深的不安全感、我们对自己不够好的确定性联系在一起。我们对阻挡入口的“天才专享”标志感到恐惧，忘记了我们把在我们告诉自己数学对我们来说太难的那一天，我们就报名了。

Fear is our real limitation. It concerns all of us, at every level, from the worst to the best, beginners to famous academics. We all have our blind spots, those words whose simple utterance fills us with terror because we’ve associated them with our deepest insecurities, our certainty that we’re not good enough. We’re petrified by the sign “Reserved for geniuses” that bars the entrance, forgetting that we put the sign up ourselves the day we told ourselves that math was too hard for us.

最令人难过的是，对数学的恐惧是，即使你知道这只是你脑子里的想法，但什么也改变不了。这就像恐高症：你知道这只是你脑子里的想法，但你还是会害怕。

The saddest thing about the fear of math is that even though you know it’s just in your head, it doesn’t change anything. It’s like fear of heights: you know it’s only in your head, but all the same, you’re still afraid.

失败的对话

Failed Conversations

在我数学进步的整个过程中，我经历了三次重大突破——三次解放时期，随着我的心理态度的改变，我感到内心的恐惧正在消退。

Throughout my progress in math, I’ve experienced three big breakthroughs—three periods of liberation when, following a change in my psychological attitude, I felt the fear in me receding.

我在第9章和第 10 章中谈到了前两个情节。首先，通过关注直觉和逻辑之间的不一致，我摆脱了第一次尝试失败的恐惧。我允许自己自由想象，即使我还没有完全理解某些事情。然后，凭借极端的心理可塑性，我开始解决对不够聪明的恐惧。如果我坦率而真诚地观察世界，如果我花时间吸收一切，就有可能克服我的局限性并变得富有创造力。

I spoke about the first two of these episodes in chapters 9 and 10. First, by paying attention to the dissonance between my intuition and logic, I chased away my fear of not succeeding on the first try. I allowed myself to imagine freely, even when I didn’t yet fully understand something. Then, betting on an extreme mental plasticity, I started to address my fear of not being smart enough. If I observed the world with candor and sincerity, if I took the time to soak it all in, it was possible to overcome my limitations and become creative.

第三次也是最意想不到的突破发生在我三十多岁的时候。我学会了摆脱害怕被认为不够聪明的恐惧。

The third and most unexpected of these breakthroughs happened later, when I was in my thirties. I learned to chase away my fear of being perceived as not smart enough.

直到那时，尽管我的职业生涯开端光荣，并取得了一些初步成功，但我仍然坚信自己不是真正的数学家。我把我的成功归功于运气。我告诉自己，我是个冒名顶替者，最终会被发现。当我在耶鲁大学任教时，我做了真正的噩梦。

Up until then, despite an honorable beginning to my career and some initial success, I remained convinced that I wasn’t a real mathematician. I attributed my success to luck. I told myself that I was an imposter, and that I would end up being found out. When I was teaching at Yale, I was having actual nightmares.

我们最深的恐惧往往是社交方面的。对于数学家来说，我们常常担心自己不如别人聪明，害怕别人会发现这一点。

Our deepest fears are often social. For mathematicians, we’re often afraid we’re not as smart as the others, and that they’ll see it.

我看到大多数年轻数学家眼中都有同样的恐惧我遇到过。这是一个很自然的现象。我在第 4 章中谈到了视觉错觉，这种错觉使我们低估了我们真正理解的数学的难度，原因很简单，它对我们来说似乎很明显。

I’ve seen this same fear in the eyes of most young mathematicians I met. It’s a natural enough phenomenon. I spoke in chapter 4 of the optical illusion that makes us underestimate the difficulty of math we really understand for the simple reason that it seems obvious to us.

第二个因素特别与年轻学者有关。通常，学者是知识渊博的人。当你成为一名职业数学家时，你的社会身份就变成了聪明人。但事实并非如此，而且没有人警告过你这一点。

There’s a second factor that specifically concerns young academics. Normally, an academic is someone who knows things. When you become a professional mathematician, your social identity becomes that of someone who is smart. Except it doesn’t at all work that way, and no one has warned you about it.

这种误解可能会导致严重的冒名顶替综合症。我知道有些人被这种误解所困扰，甚至永久性地损害了他们的创造力。

This misunderstanding can give rise to an aggressive form of imposter syndrome. I know people who have been overwhelmed by it, to the point where it permanently damaged their creativity.

数学是一种实践，而不是知识。数学家比任何人都更了解他们所研究的对象，但他们的数学直觉永远无法变得无所不能。他们不熟悉的对象仍然会带来困难。你可以是一名出色的运动员，奥运会标枪冠军，身体状况处于巅峰状态，但这并不能阻止你在网球比赛中被一名优秀的青少年球员击败。

Mathematics is a practice rather than knowledge. Mathematicians understand better than anyone the objects they’re working on, but their mathematical intuition can never become omnipotent. Objects they aren’t familiar with still raise difficulties. You can be an exceptional athlete, Olympic champion with the javelin, in peak physical condition, but that won’t stop you from being crushed at tennis by a decent junior player.

在数学研究中，没有权威地位。这造成了令人不安、情绪困扰的情况，违背了社会对我们的期望。

In mathematical research, there is no position of authority. That creates disturbing, emotionally troubled situations that go against what is socially expected of us.

这是我亲身经历过的一个现实情况。你是一名聪明的年轻研究员。你刚刚获得一个享有盛誉的职位，并被邀请在一次国际会议上发表演讲。晚餐时，你发现自己坐在一位年轻的博士生旁边，她正在解释她正在研究的内容。你一句话都听不懂。你问了她一个问题。你不明白她的回答。因为你很固执，所以你冒险直接告诉她你不明白。“别担心，”她说，“让我用一个简单的例子重新解释一下；你马上就会明白。”然而她用不同的词重新解释了一遍，而你仍然不明白她在说什么。

Here’s a real-life situation that I personally experienced. You’re supposed to be a bright young researcher. You’ve just gotten a prestigious position and you’re an invited speaker at an international conference. At dinner, you find yourself seated next to a young doctoral student who is explaining what she’s working on. You don’t understand a word she’s saying. You ask her a question. You don’t understand her answer. Since you’re stubborn, you risk telling her straight out that you don’t understand. “No worry,” she says, “let me reexplain it with a simple example; you’ll get it right away.” And there she reexplains it using different words, and you still don’t understand anything she’s saying.

问题不在于她的解释，而在于你。要想理解，你必须从头开始，从基础开始。她的工作与一个你应该在研究生院学到的理论有关，但你却从未理解过。当然，你也不敢告诉任何人。

The problem isn’t with her explanation. The problem is with you. To understand, you’d have to start all over from the beginning, from the basics. Her work is related to a theory that you’re supposed to have learned in graduate school, but which never made any sense to you. And, of course, you never dared tell anyone.

你处在社会可接受范围的边缘。你的信誉岌岌可危。如果你承认自己有多迷失，你会看起来像个傻瓜。社会规范是放手。

You’re at the edge of what’s socially acceptable. Your credibility is at stake. If you acknowledge how lost you are, you’ll look like a fool. The social norm is to let it go.

这种情况是所有失败的数学对话的典型情况，我们从中得不到任何教训，只能强化我们认为自己是最差的那个。

This situation is typical of all failed math conversations, those we learn nothing from, those that serve only to reinforce our certainty that we’re the worst of the worst.

无论你的数学水平如何，你都知道我在说什么。绝大多数数学对话都以这种萎靡不振的感觉结束。它们失败的原因很简单：你不敢说你有多迷茫。你感到羞愧，你觉得荒谬，这种想法折磨着你，让你无法倾听。你只想到自己的无用。这就是阻止你想象和学习的原因。你从这些对话中感到羞辱。

Whatever your level of math, you know what I’m talking about. The vast majority of math conversations end with this feeling of malaise. They fail for this simple reason: you don’t dare say how lost you are. You’re ashamed, you feel ridiculous, and this idea gnaws at you and makes you incapable of listening. You think only of your own worthlessness. It’s what keeps you from imagining and learning. You come out of these conversations feeling humiliated.

当我三十二岁时，我学会了一种社会工程技术来改变这些对话的动态。

When I was thirty-two, I learned a social engineering technique to change the dynamics of these conversations.

我从让-皮埃尔·塞尔 (Jean-Pierre Serre) 那里直接学到了这项技巧，我们在第 7 章中提到过他，格罗滕迪克曾写信给他谈论他的“荒谬的作品”。

I learned the technique directly from Jean-Pierre Serre, whom we spoke of in chapter 7, the person Grothendieck wrote to about his “ridiculous piece.”

这堂课只有五秒钟，只有一句话。这是我一生中上过的最有效的数学心理学课。我思考了几个月才完全理解了它的范围。多亏了这种方法，我再也没有在数学对话后感到丢脸。

The lesson lasted five seconds flat and consisted of a single sentence. It’s the most effective lesson in mathematical psychology that I’ve had in my life. I thought it over for months before I completely understood its scope. Thanks to this method, I’ve never again left a math conversation feeling humiliated.

我怀疑其中有因果关系：在三十二岁到三十五岁之间，我的数学能力得到了极大的提升。理解。我第一次感到完全合法和安心。我在研究中取得了惊人的进展，我对这段时间证明的定理感到非常自豪。

I suspect there’s a cause and effect: between age thirty-two and thirty-five, I underwent a formidable acceleration of my mathematical understanding. For the first time, I felt completely legitimate and at ease. I made spectacular progress in my research and I’m quite proud of the theorems I proved during this period.

数学演讲的艺术

The Art of Giving Math Talks

我当然会分享 Serre 教给我的技术，但首先我必须解释一下背景。

I’ll of course share the technique Serre taught me, but first I have to explain the context.

在新的数学成果被编辑和发表之前，它们通常以口头形式在研讨会和会议上发表。我一直很喜欢做数学演讲，尽管这种形式可能令人生畏，尤其是在黑板前。演讲通常持续一个小时，你一个人拿着一支粉笔，面对一群专家，他们冷漠地看着你，随时可能打断你提问。这没有留下太多虚张声势的余地。但这正是令人兴奋的地方。

Before new mathematical results are edited and published, they’re generally presented in oral form at seminars and conferences. I’ve always enjoyed giving math talks, although the format can be intimidating, especially when it’s at the blackboard. Talks usually last an hour, you’re all alone with a piece of chalk in your hand, facing an audience of specialists who look at you impassively and may at any time interrupt you with questions. It doesn’t leave a lot of room to bluff. But that’s what’s exciting.

我清楚地记得 1997 年在剑桥大学艾萨克·牛顿数学科学研究所的第一次研究报告。当时我是一名博士生，非常害怕。为了克服自己的不安全感，我选择将演讲定位在最基本的层面，这需要大量的准备。我寻找用最简单的心理图像和最自然的联系来讲述故事的方法。

I remember very well my first research presentation, in 1997 at the Isaac Newton Institute for Mathematical Sciences in Cambridge. I was a PhD student and I was absolutely terrified. To fight against my own insecurities, I chose to position the talk at the most elementary level possible, which demanded an incredible amount of preparation. I looked for ways to tell the story with the simplest mental images and the most natural connections.

在某种程度上，我试图尽量减少精神能量——包括我自己的和观众的精神能量。一个很好的比喻是攀岩：要爬上悬崖，你需要找到一条路径和一系列需要最少努力的动作。只有当事情很容易时，你才能成功。如果很难，你最终会伤害到自己。

In a way, I was trying to minimize mental energy—my own as well as that of the audience. A good metaphor is rock climbing: to climb up a cliff, you need to find a path and a series of actions that require minimal effort. You can succeed only if it’s easy. If it’s hard, you’ll end up hurting yourself very, very badly.

这次演讲对我来说是一个启示。我明白，只有通过向别人解释，我才能真正理解自己的结果。这是一个众所周知的现象，数学家们有一句话说，数学课唯一的用处就是让教授听懂。

This talk was a revelation for me. I understood that it was only by explaining it to others that I was able to really understand my own results. This is a well-known phenomenon, and mathematicians have a saying, that the only thing a math lesson is good for is to allow the professor to understand.

对我来说，理解自己的数学知识的最好方式就是想象自己必须向完全的初学者解释数学知识。通过自欺欺人，我最终找到了将结果呈现得显而易见的方法。

The best way for me to understand my own math is to imagine that I have to explain it to complete beginners. By playing the fool with myself, I end up finding ways to present my results as being obvious.

这种简约的风格成为了我的演讲风格，与许多年轻数学家喜欢隐藏的深奥风格和夸夸其谈截然相反。我最初担心我的演讲幼稚无知不会给我带来任何好处。风险在于人们不会认真对待我。但事实恰恰相反。我的演讲越简单，人们就越认为我聪明。

This minimalist approach became my presentation style, as opposed to the esoteric style and bluster behind which a lot of young mathematicians like to hide. I was initially afraid that the naïveté of my presentations wasn’t doing me any favors. The risk was that people wouldn’t take me seriously. The opposite happened. The simpler my talks were, the more intelligent people thought I was.

有一天，我必须在巴黎的 Chevalley Seminar 上做演讲，这是一个群论研讨会。虽然我没有实质性的新成果可以宣布，但这是一个比平时更简单的演讲机会。

One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. I didn’t have substantial new results to announce, but it was an opportunity to make a presentation even simpler than usual.

我到达房间时，那里有大约十五名研究人员，还有几名学生坐在后面。在演讲开始前几分钟，塞雷进来坐在第二排。

When I got to the room, fifteen or so researchers were there, along with a few students seated in the rear. A couple of minutes before the talk was to start, Serre came in and sat in the second row.

我很荣幸他能成为我的听众，但我马上就告诉他，他可能对这次演讲不太感兴趣。这次演讲面向普通听众，我只会解释一些非常基础的东西。

I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things.

当然，我没有告诉他的是，他的出现令人生畏。不过，我不想仅仅为了引起他的兴趣而提高我的演讲水平。我只是留意他是否摘下了眼镜，这意味着他感到无聊并且不再听了。不用担心——他一直戴着眼镜直到最后。

What I didn’t tell him, of course, was that his presence was intimidating. Still, I didn’t want to raise the level of my talk only to keep him interested. I just kept an eye out to see if he’d taken off his glasses, which would mean he was getting bored and had stopped listening. No worries there—he kept his glasses on till the end.

我像没有他在场一样进行了演讲，向全体听众，特别是坐在后面的学生讲话，我很高兴看到他们在听，而且看起来好像他们明白了。

I gave my presentation as I would have without him there, speaking to the entire audience, especially the students seated in the back, whom I was pleased to see listening and looking like they understood.

这是一次普通的演讲，相当成功，虽然不是很深入，但准备充分，思路清晰，易于理解。研讨会结束时，Serre 走到我面前说——我在此逐字引用：“你得再给我解释一遍，因为我什么都听不懂。”

It was a normal presentation, fairly successful, not very deep but well prepared, clear, and intelligible. At the end of the seminar, Serre came up to me and said—and here I quote verbatim: “You’ll have to explain that to me again, because I didn’t understand anything.”

看起来像个傻瓜

Looking Like a Fool

这是一个真实的故事，它让我陷入深深的困惑之中。

That’s a true story, and it plunged me into a state of profound perplexity.

显然，Serre 并没有理解大多数人使用动词的方式。我演讲中的概念和推理实际上不会给他带来任何困难。我相信他想说他理解了我所解释的内容，但他不明白我所解释的内容为什么是正确的。

Apparently, Serre wasn’t using the verb to understand the way most people use it. The concepts and reasonings of my talk couldn’t really have caused him any difficulty. I’m sure he wanted to say that he understood what I had explained, but he hadn’t understood why what I had explained was true.

这有点像 1 到 100 的整数之和，理解有两个层次。第一个层次是一步步遵循推理并接受它是正确的。接受并不等同于理解。第二个层次是真正的理解。它需要看到推理从何而来，以及为什么它是自然的。

It’s a bit like with the sum of whole numbers from 1 to 100, where there are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it’s correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it’s natural.

再次思考 Serre 的评论时，我意识到我的演讲有太多“奇迹”，太多随意的选择，太多事情在我不知道原因的情况下就成功了。Serre 是对的；这太不可思议了。他的反馈帮助我意识到，我对当时正在研究的对象和情况的理解存在许多非常大的漏洞。

In thinking again about Serre’s comment, I realized that my presentation had too many “miracles,” too many arbitrary choices, too many things that worked without my really knowing why. Serre was right; it was incomprehensible. His feedback helped me become aware of a number of very big holes in my understanding of the objects and situations I was working on at the time.

在接下来的几年里，对这些不同奇迹的解释的研究让我填补了一些空白，并取得了职业生涯中最重要的一些成果。（然而，有些奇迹至今仍未得到解释。）

In the years that followed, research into explanations for these various miracles allowed me to fill in some of the holes and achieve some of the most important results of my career. (However, some of the miracles remain unexplained to this day.)

但最令人不安的还是塞尔的唐突和坦率，他过分强调了自己的无知。

But the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension.

仔细聆听演讲，然后走到演讲者面前，微笑着告诉他你“什么都不懂”，这需要很大的勇气。我从来都不敢这么做。

It takes a lot of nerve to listen closely to a presentation, then go up to the speaker, smile, and tell him that you “didn’t understand anything.” I never would have dared.

他为什么要这么做？我首先告诉自己，当你是让-皮埃尔·塞尔时，这一定是你有权做的事情之一。然后我意识到这也可以反过来：如果这种技术真的帮助他成为了让-皮埃尔·塞尔，那会怎样？

Why did he do it? I first told myself it must be one of the things you have the right to do when you’re Jean-Pierre Serre. Then I realized that could also work the other way: what if this technique had actually helped him become Jean-Pierre Serre?

为了确保万无一失，我决定亲自尝试一下。

I decided to try it myself, just to be sure.

几个月后，在一次会议上，我发现自己坐在一位博士生旁边。吃甜点的时候，他开始跟我谈论他正在研究的内容。不用说，我一句话都听不懂。晚饭后，我把他拉到一边说：“再给我解释一遍，但要简单、慢慢地讲。我对你的课题一无所知。假设我有脑损伤，无法集中注意力超过几秒钟。”

A few months later, at a conference, I found myself at a table next to a PhD student. During dessert, he started talking to me about what he was working on. Needless to say, I didn’t understand a word he was saying. After dinner, I took him aside and said: “Explain it to me again, but very simply, very slowly. I don’t understand anything about your subject. Assume that I have brain damage and can’t focus my attention for more than a few seconds.”

这让他笑了，并且他很好心地从头开始，慢慢地、平静地向我解释他所在领域的基础知识，这些我应该知道，但到目前为止还没有理解。

That made him smile, and he had the kindness to explain it slowly and calmly, starting from the beginning, with the basics in his field, that I should have known but up till now had never succeeded in understanding.

他的解释和晚餐时说的完全不一样。他使用的词语不一样，甚至谈论的也不是同一个东西。就好像他对于研究课题有两种截然不同的解释方式。就好像有一份旅游菜单，是他为了显得严肃而提供的官方解释；还有一份秘密菜单，是他自己理解事物的简单而直观的方式。

His explanation didn’t have anything to do with what he had said over dinner. He didn’t use the same words, and didn’t even talk of the same things. It was as if he had two completely different ways of talking about the subject of his research. It was like there was the tourist menu, the official explanation he served up when he wanted to appear serious, and the secret menu, the simple and intuitive way he understood the things himself.

因为他是学生，而我是资深学者，所以我的社会地位更高。晚餐时，他为了给我留下好印象，给我端来了游客菜单。通过夸大自己的无用，我让他把自己放在平等的地位，然后脱口而出。

Because he was a student and I was an established scholar, I had higher social status. At dinner, he had tried to impress me by serving me the tourist menu. By overplaying my own worthlessness, I granted him permission to place himself on equal footing and just blurt it out.

Serre 技术的另一个好处是它能轻易去除您想问的所有愚蠢问题的戏剧性。与其一点一点地问这些问题，感觉自己在谈话中每过一刻都在倒退，失去尊严，不如从一开始就直接进入话题，承认您肯定会问很多愚蠢的问题，甚至会一遍又一遍地问同样的愚蠢问题，这样会更舒服。

Another benefit of Serre’s technique is that it readily takes away the drama of all the stupid questions that you’d want to ask. Instead of asking them bit by bit, having the feeling that you’re going backwards and losing your dignity with each passing moment of the conversation, it’s much more comfortable to jump right in from the outset, admitting that sure, you’re going to ask a lot of stupid questions, and you’ll even be asking the same stupid questions over and over.

如果你开始一场关于数学的对话，是为了学习一些东西，而不是为了受羞辱。

If you start up a math conversation, it’s to learn something, not to be humiliated.

有时你会花一半的时间来复习你误解的基础知识，有时你只做这些。无论如何，这比谈论你无法理解的事情要好。如果你交谈的人没有把自己放在你的水平上，拒绝从基础开始，手把手地指导你，那么心烦意乱也是没有用的。你可能偶然发现了一个真正的骗子，有人假装解释超出他们自己理解范围的数学。真正的冒名顶替者是那些没有这种综合症的人。

Sometimes you spend half the time reviewing the basics that you’d misunderstood, and sometimes that’s all you do. At any rate, that’s better than to talk about things you can’t make any sense of. If the person you’re talking with doesn’t place themself at your level, and refuses to start with the basics and lead you by the hand, there’s no use getting distraught. You’ve probably stumbled across an actual fraud, someone who pretends to explain math that is beyond their own comprehension. The real imposters are the ones without the syndrome.

这种方法的妙处在于，通过扮演傻子，你最终会以自己的自信给别人留下深刻印象。

The beauty of this approach is that by playing the fool you’ll end up impressing people with your own self-confidence.

拒绝恐惧

Refusing Fear

Serre 的技巧简单而有效。而且似乎每个人都能做到。理论上，没有什么能阻止你看着别人的眼睛，笑着告诉他们你不明白某件事，他们需要从头再解释一遍。显然，这不是智商的问题。

Serre’s technique is simple and powerful. And it would seem everyone could do it. In theory, nothing is stopping you from looking people in the eyes and telling them smilingly that you didn’t understand a thing and they need to explain it again from the beginning. Clearly, it’s not a question of IQ.

尝试一下你就会明白。

Try it and you’ll see.

这看似简单，实则不然。虚张声势并假装自己明白可能很难。更难的是完全停止虚张声势，不加筛选地、不加思索地问出所有浮现在脑海中的愚蠢问题。羞耻。Serre 的技术是我们在第 7 章中所说的 儿童姿势的社交版本。这需要很好地控制你的身体和情绪，因为我们有隐藏无知的本能。

It seems easy, but it’s not. It might be hard to bluff and pretend that you understand. It’s even harder to stop bluffing altogether and ask all the stupid questions that come to mind, without filter, without shame. Serre’s technique is the social version of what we called in chapter 7 the child’s pose. This requires a great command of your body and emotions, because we have the instinct to hide our ignorance.

如果你担心自己会被当成一个真正的白痴，那么数学会变得更加困难。这就是为什么数学会放大所有的刻板印象和社会不安全感。如果你属于一个缺乏认可和榜样的少数群体，如果你暗自相信你的基因使你无法理解数学，或者只是数学不好已经成为你社会身份的一部分，那么对你来说，数学会变得困难得多。

And it’s spectacularly harder if you worry that you might be written off as an actual idiot. This is why math is such an amplifier for all stereotypes and social insecurities. If you’re part of a minority that lacks recognition and role models, if you’re secretly convinced that your genes make you incapable of understanding math, or simply if being bad at math has become part of your social identity, it will be so much harder for you.

最后，担心数学太难是一种自我实现的预言。除了一些实用技巧和继续努力的一般建议外，我没有简单的解决办法。

In the end, fearing that math is too difficult for you is a self-fulfilling prophecy. I have no easy fix to offer, apart from a few practical tips and the generic advice to keep pushing.

塞尔教会我，直截了当比拐弯抹角要好。你不妨开玩笑地揭露让你感到羞耻的事情和你想隐藏的事情。幽默是我所知道的对抗恐惧的最佳武器。通过将自己的智力限制推向荒谬的水平，你可以创建一个临时的童真自由区，在那里允许提出任何和所有问题。

What Serre taught me was that it’s better to be straightforward and direct rather than to beat around the bush. You might as well make fun of revealing what makes you ashamed and what you’d like to hide. Humor is the best weapon I know against fear. By pushing your own intellectual limitations to absurd levels, you can create a temporary zone of childlike freedom where any and all questions are allowed.

寻找合适的导师也很重要。我已经在皮埃尔·德利涅获得阿贝尔奖后接受的视频采访的第 9 章中谈到了这一点。这是他分享数学愿景并回顾职业生涯中一些决定性时刻的机会，包括他与格罗滕迪克的第一次互动，后者后来成为他的博士导师。

It’s also essential to seek the right mentors. I’ve already spoken in chapter 9 of the video interview given by Pierre Deligne after he received the Abel Prize. It was an opportunity for him to share his vision of mathematics and recount some of the decisive moments of his career, including his first interaction with Grothendieck, before the latter became his PhD advisor.

当时还是一名年轻学生的德利涅参加了格罗滕迪克的一次研讨会，被他高大的身影和光头吓到了。在演讲过程中，格罗滕迪克滔滔不绝地谈论着“上同调对象”，这是他工作中的一个核心数学概念。但德利涅一点也听不懂。在研讨会结束时讲座中，他找到格罗滕迪克并请他解释“上同调对象”的含义。

Deligne, who was a young student at the time, went to a seminar given by Grothendieck, and was intimidated by his large silhouette and shaved head. During the presentation, Grothendieck spoke nonstop about “cohomology objects,” a mathematical concept central to his work. But Deligne didn’t understand any of it. At the end of the lecture he approached Grothendieck and asked him to explain what he meant by “cohomology objects.”

这有点像听完爱因斯坦的讲座后，走到他面前问他“相对论”是什么意思。半个多世纪后，德利涅仍然钦佩格罗滕迪克的反应：“他当时的反应很典型。其他人会认为，如果我不知道这是什么，那么真的不值得和我说话。但他的反应完全不是这样。他非常耐心地向我解释。”

It’s a little like sitting through a lecture by Einstein and going up to him afterwards to ask what he meant by “relativity.” More than a half century later, Deligne still admired Grothendieck’s reaction: “That was really typical of him. Other people would have thought that, if I didn’t know what this was, then really it was not worth speaking with me. That was not his reaction at all. Very patiently he [explained it to me].”

这种耐心和仁慈对德利涅产生了很大的影响，并使他得以蓬勃发展：

This patience and benevolence had a great impact on Deligne, and allowed him to flourish:

他非常和蔼，可以问一些看似完全愚蠢的问题。和他在一起，我一点也不害羞，问一些看似完全愚蠢的问题，这个习惯一直保持到现在。我通常坐在听众前面听讲座，如果我有什么不明白的地方，我会问，即使我应该知道答案是什么。

He was extremely kind, one could ask apparently completely stupid questions. Being with him, I wasn’t shy at all asking questions which would be completely stupid, and I’ve kept this habit until now. I usually sit in front of the audience attending a lecture, and if I have something I don’t understand I’ll ask, even if I would be supposed to know what the answer is.

这些并非只是说说而已。如果德利涅花时间坚持下去，那是因为他知道这有多难。他看到很多数学家正是在这一点上失败了，因为他们无法达到正确的坦率和透明度。数学中最困难的事情是克服我们的羞耻感或逃避的本能，以及我们掩饰的本能。这都是一个镇定和身体参与的问题。

These aren’t just words. If Deligne takes the time to insist, it’s because he knows how difficult it is. He’s seen so many mathematicians fail at precisely this point, because they aren’t able to attain the right level of candor and transparency. The most difficult thing in math is to overcome our shame or instinct for flight, our reflex for dissimulation. It’s all a question of composure and physical engagement.

瑟斯顿在 MathOverflow 的社交媒体资料中写了类似的话：

In his social media profile on MathOverflow, Thurston wrote something similar:

数学是一个过程，需要你用足够的毅力，用足够的努力去面对混乱和困惑的迷雾，最终突破迷雾，获得更清晰的思路。当我至少对自己承认我的思维混乱时，我会很高兴，我会努力克服这种混乱。担心自己可能会暴露自己的无知或困惑。多年来，这帮助我在某些事情上变得清晰起来，但在其他很多事情上我仍然一头雾水。

Mathematics is a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others.

塞尔、德利涅、瑟斯顿、格罗滕迪克：如果这些杰出的数学家都坚持同一条观点，那绝非巧合。与我们的禁忌和障碍作斗争是数学工作的本质。

Serre, Deligne, Thurston, Grothendieck: if these outstanding mathematicians all insist on the same point, it’s not a coincidence. The struggle against our inhibitions and stumbling blocks is the essence of mathematical work.

不理解是正常的。害怕是正常的。必须努力抑制恐惧也是正常的。事实上，这正是危在旦夕的事情。

It’s normal not to understand. It’s normal to be afraid. It’s normal to have to struggle to contain your fear. It is, in fact, precisely what’s at stake.

14

武术

14

A Martial Art

1649 年初，勒内·笛卡尔通过法国驻斯德哥尔摩大使收到了瑞典女王克里斯蒂娜的邀请，她希望他能来给她上私人课。

Early in the year 1649 René Descartes received an invitation from Queen Christine of Sweden via the ambassador of France at Stockholm. She wanted him to come and give her private lessons.

在接受邀请之前，笛卡尔想确认女王是认真的。他告诉大使：如果这只是一时兴起，如果女王没有真正学习的必要动力，他就不会来。

Before accepting, Descartes wanted to make sure that she was serious. He told the ambassador: if this is only a whim, if the queen doesn’t have the necessary motivation to truly learn, he wouldn’t make the trip.

前一年，他接受了巴黎的邀请，浪费了自己的时间：“最让我反感的是，他们除了我的脸之外，没有一个人表现出想要了解我的任何其他事情的欲望。”这就是名人的代价。他觉得人们​​想要他不是因为他的想法，而是“像大象或豹子一样，因为它很稀有。”

The year before, he’d wasted his time by accepting an invitation to Paris: “What disgusted me the most was that none of them showed any desire to know anything else about me but my face.” It is the price of celebrity. He had the feeling that people wanted him not for his ideas, but “like an elephant or a panther, because of its rarity.”

笛卡尔比爱因斯坦早三个世纪，是第一批获得摇滚明星地位的知识分子之一。

Three centuries before Einstein, Descartes was one of the first intellectuals to achieve the status of a rock star.

他最终接受了克里斯蒂娜女王的邀请，前往斯德哥尔摩，并于 1650 年 2 月 11 日因肺炎去世，享年 53 岁。在将他的遗体运回法国时，他的头骨被盗。它在黑市上流通了两个世纪。不同的主人在上面刻上自己的名字，仿佛头骨被赋予了他们可以占有的神奇力量。头骨最终在巴黎人类博物馆安息，与南方古猿的头骨一起展出。

He ended up accepting Queen Christine’s invitation and went to Stockholm, where he died of pneumonia at the age of fifty-three on February 11, 1650. When his body was being returned to France, his skull was stolen. It circulated on the black market for two centuries. The various owners engraved their names on it, as if the skull were endowed with magical powers that they could appropriate. The skull finally found its resting place in the Musée de l’Homme in Paris, where it is displayed next to the skull of an Australopithecus.

“世界上分布最均匀的东西”

“The most evenly distributed thing in the world”

现在，我们已经熟记这个故事。这个故事和我们从第一页开始就不断重复的故事是一样的。这个故事讲述了我们拒绝相信数学首先是一种态度，我们坚持认为擅长数学的人一定是脑子有某种异常。当一个真正的天才敢说不是这样的时候，我们会切开他的头看看里面有什么。

By now, we know the story by heart. It’s the same story we’ve been repeating from page 1. It’s the story of our refusal to believe that math is first and foremost an attitude, and our insistence that people who are good at it must have some kind of brain abnormality. And when an actual genius dares say otherwise, we cut his head open to see what’s inside.

1637 年，在成为名人之前，笛卡尔发表了一篇自传体文章《方法论》，其中概述了他的思想历程。他揭示了自己的工作方法，并讲述了他如何成为当时最伟大的数学家。从开篇开始，信息就非常明确，笛卡尔认为他没有什么特殊的天赋：

In 1637, before becoming a celebrity, Descartes had published an autobiographical essay, Discourse on Method, in which he outlined his intellectual journey. He revealed his work methods and told how he had become the greatest mathematician of his time. From the opening lines, the message is radically clear, and Descartes thinks he has no special talent:

就我个人而言，我从未认为自己的思维比普通人更发达。事实上，我常常希望我的思维和其他人一样敏捷，我的想象力和其他人一样清晰准确，我的记忆力和其他人一样丰富敏锐。

For myself, I have never presumed my mind to be any way more accomplished than that of the common man. Indeed, I have often wished that my mind was as fast, my imagination as clear and precise, and my memory as well stocked and sharp as those of certain other people.

笛卡尔承认，他有幸偶然发现了一种特殊的方法，让他以不同的方式看待事物：

What Descartes admits to is having a different way of looking at things, thanks to a particular method that he was lucky enough to stumble upon:

我已经找到了一种方法，在我看来，通过这种方法，我可以逐步增加我的知识，并逐渐将其提高到我心智的局限性和我短暂的生命所允许的最高点。

I have fashioned a method by which, it seems to me, I have a way of adding progressively to my knowledge and raising it by degrees to the highest point that the limitations of my mind and the short span of life allotted to me will permit it to reach.

这种方法，正如他向我们描述的，简单得像个孩子。它只需要一种精神资源，即“良好的意识”，即我们所有人都赋予。换句话说，我们都有成为笛卡尔的潜力。为了不留下任何疑问，他在书的开头用一句口号的形式写道：“良好的判断力是世界上最均匀分布的东西。”

This method, as he describes it to us, is of a childish simplicity. It requires only one mental resource, the “good sense” that we’re all endowed with. In other words, we all have the potential to become Descartes. In order not to leave any doubt, he opens his book with a sentence in the form of a slogan: “Good sense is the most evenly distributed thing in the world.”

我们从来没有机会与爱因斯坦进行讨论。《收获与播种》是一本晦涩难懂、几乎无法阅读的文本。另一方面， 《方法论》是思想史上阅读和评论最广泛的文本之一。为什么在它出版近四个世纪后，几乎没有人知道存在一种提高数学水平的方法？

We never had the chance to have our discussion with Einstein. Harvests and Sowings is an obscure, almost unreadable text. Discourse on Method, on the other hand, is one of the most widely read and commented upon texts in the history of thought. How is it possible that nearly four centuries after its publication almost no one is aware of the existence of a method for becoming good at math?

坦白说，我们集体无法读懂笛卡尔的作品，这很不寻常。我们假装读过它，假装理解它，假装认为它很重要，但实际上我们坚决拒绝认真对待它。在内心深处，我们都相信他在嘲笑我们。

Our collective inability to read Descartes is quite frankly extraordinary. We pretend to read it, we pretend to understand it, we pretend to find it important, but in reality we categorically refuse to take it seriously. Deep down we’re all convinced that he’s making fun of us.

这种误解根深蒂固，也在很大程度上解释了我们未能实现数学民主化的原因。但其影响远不止于此。事实上，其严重程度令人发指。

This misunderstanding runs deep and explains much of our failure to democratize mathematics. But it is much broader than that. In fact, its extent is abysmal.

事实上，笛卡尔并没有止步于数学。在发展了自己的独特方法、以铁的纪律实践了它、通过伟大的数学发现验证了它之后，他开始用它来重建整个科学和哲学。

Indeed, Descartes didn’t stop at mathematics. After developing his particular method, after practicing it with an iron discipline, after validating it through great mathematical discoveries, he set out to use it to reconstruct the entirety of science and philosophy.

他创立的思想流派被称为理性主义。我们的科学技术就是其直接的后代。理性主义遇到了笛卡尔未曾预见到的局限和陷阱，我们稍后会谈到这一点。但这并没有影响它的成功。理性主义方法在我们每个人身上都存在，无论我们喜欢与否，我们都知道理性是有目的的，就像我们都知道数学非常强大一样。即使我们很难解释为什么。

The school of thought he founded is called rationalism. Our science and technology are direct descendants. Rationalism encountered limitations and pitfalls that Descartes didn’t foresee, which we’ll return to later. That doesn’t take away from its success. The rationalist approach lives on in each of us, and whether we like it or not, we all know that rationality serves a purpose, just as we all know that math is extraordinarily powerful. Even if we have a hard time explaining why.

当我们拒绝认真对待笛卡尔时，我们拒绝理解理性本身。

When we refuse to take Descartes seriously, it’s rationality itself that we are refusing to understand.

《方法论》不是一本理论书。这是一份个人证言，笛卡尔在其中描述了他自己试验过的一些心理技巧。他断言这些技巧使他能够发展自己的认知能力、建立自信并做出伟大的发现。为了证明他的观点，他在书中附上了三篇科学文献，其中包括《几何学》，这是一部具有革命性的数学著作，它重塑了我们的语言和想象力（笛卡尔在历史上第一次用字母x来表示未知数）。

Discourse on Method isn’t a book of theory. It’s a personal testimony, in which Descartes describes a number of mental techniques that he experimented with on himself. He affirms that these techniques allowed him to develop his cognitive abilities, build self-confidence, and make great discoveries. As proof of what he’s saying, he accompanies his text with three scientific texts, including Geometry, a mathematical work so revolutionary that it remade our language and imagination (for the first time in history, Descartes used the letter x to designate an unknown).

《方法论》是一本自助书籍，其信息很简单：我们有能力构建自己的智慧和自信。

Discourse on Method is a self-help book whose message is simple: we have the ability to construct our own intelligence and self-confidence.

秘密的合理性

The Secret Rationality

当你是一名数学家时，你经常会遇到这样的人：“你是一个理性的人”或“你是一个喜欢逻辑的人”，或者（最糟糕的是）“你是一个擅长数学的人”。

When you’re a mathematician, you often meet people who say to you, “You’re the rational one” or “You’re the one who likes logic” or (worst of all) “You’re the one who’s good at math.”

这始终是一个不好的迹象，因为在这背后总是隐含着这样一种语气：“你就是个怪人”，“你对生活一无所知”，或者“我要向你倾诉我在学生时代积累的所有挫败感”。

It’s always a bad sign, because behind that there is always an underlying tone of “You’re one of those weirdos,” “You know nothing about life,” or “You’re the one on whom I’m going to pour out all the frustration I’ve built up through my school years.”

理性的名声和数学一样糟糕。与数学一样，理性也有两种形式。理性的可见一面表现为既定的知识、科学技术以及结构良好、逻辑合理的论证。学校花了很多时间教授理性，但效果好坏参半。

Rationality has as bad a reputation as mathematics. And like the latter, it exists in two versions. The visible side of rationality presents itself in the form of established knowledge, science and technology, and well-structured and logically sound arguments. Schools spend a lot of time teaching it, with mixed results.

理性的另一面，即它的秘密和亲密维度，仍然基本上没有被记录下来，就好像我们故意选择掩盖它一样。

The flip side of rationality, its secret and intimate dimension, remains largely undocumented, as if we’d made a deliberate choice to cover it up.

在第 11 章中，我提出了理性（以及合理性）作为同义词系统 2（遵循规则和逻辑的机械思维）的缩写。这是一条方便的捷径，因为对很多人来说，这就是它的含义。但它也会导致重大问题。通过将理性与系统 1（直觉和即时思维）对立起来，我们将其置于人类理解的对立面。因此，许多人认为它枯燥无味也就不足为奇了。

In chapter 11, I presented reason (and thus rationality) as a synonym for System 2 (mechanical thinking that follows rules and logic). It’s a convenient shortcut because that’s what it means to a lot of people. But it also results in major issues. By opposing rationality to System 1 (intuitive and instantaneous thinking), we’re setting it in opposition to human understanding. It’s no surprise then that so many people view it as dry and unappealing.

这种版本的理性很难让人接受，但这并不能阻止一些人试图推销它。他们往往采取一种优越和蔑视的姿态。“保持理性”是他们告诉我们“吃你的蔬菜”、“做你的家庭作业”、“尊重权威”、“抑制你的欲望”、“同意我的观点”的方式。他们命令我们保持理性，但他们无法准确解释理性的含义。他们赞美笛卡尔，却没有意识到自己对他的误解有多深。他们甚至从未想过，有了这些可怜的想法，笛卡尔早就被遗忘了，他的同时代人中没有人会像“大象或豹子”一样崇拜他。

This version of rationality is a tough sell, but it doesn’t stop some from trying to sell it. Most often they adopt a stance of superiority and disdain. “Be rational” is their way of telling us “Eat your vegetables,” “Do your homework,” “Respect authority,” “Curb your desires,” “Agree with me.” They order us to be rational but they’re incapable of explaining precisely what that consists of. They praise Descartes without realizing how much they’re getting him wrong. It never even occurs to them that, with such pitiful ideas, Descartes would have long been forgotten, and none of his contemporaries would have admired him “like an elephant or a panther.”

如果您打开《方法论》寻找对系统 2 的赞美，那么您将大失所望。笛卡尔的伟大创新是将直觉和主观性置于其知识方法的核心。他不信任既定知识和书中所写的内容。他不相信权威。他更喜欢自己在脑海中重建一切。他的方法与爱因斯坦、瑟斯顿和格罗滕迪克的方法非常相似。当然，这是系统 3，是直觉和逻辑之间缓慢而谨慎的对话，目的是发展您的直觉。

If you open Discourse on Method looking for a glorification of System 2, you’ll be sadly disappointed. Descartes’s great innovation was to put intuition and subjectivity at the heart of his approach to knowledge. He was distrustful of established knowledge and what was written in books. He placed little credit in authorities. He preferred to reconstruct everything by himself, in his head. His method closely resembles that of Einstein, Thurston, and Grothendieck. It is, of course, System 3, the slow and careful dialogue between intuition and logic, with the aim of developing your intuition.

笛卡尔对此直言不讳：他的方法只是数学家的方法。他描述它时从未提及理性或理性主义。这些词在他那个时代甚至都不存在。它们是后来发明的，用来描述他的方法。至于今天笛卡尔是否会被视为“理性的人”，我让你自己判断。

Descartes was candid about it: his method was simply that of mathematicians. He described it without ever speaking of rationality or rationalism. These words didn’t even exist in his time. They were invented later to characterize his approach. As for whether or not Descartes would pass today for a “rational person,” I’ll let you judge for yourself.

“世界上的伟大著作”

“The great book of the world”

1596 年，勒内·笛卡尔出生在法国中部的一个小村庄，后来，村里的人把这个小村庄改名为笛卡尔。笛卡尔生活的世界与我们的生活截然不同。要理解他的思想，我们必须首先了解他的世界。

René Descartes was born in 1596 in a small village in the center of France, whose inhabitants have since renamed it Descartes. The world Descartes lived in was very different from our own. In order to understand his thought, we must first understand his world.

毁掉《方法论》的最好方法就是把它视为一部世界经典，它可能会为你提供如何在学校取得成功的建议，或者一部学者的作品，它解释了如何进行研究以赢得同事的尊重。

The best way to ruin Discourse on Method is to see it as a world classic that might give you advice on how to succeed in school, or the work of an academic that explains how to do research that will win the esteem of your colleagues.

这不是笛卡尔的本意，他的生活也不符合任何现代刻板的知识分子生活。他从未担任过学术职位，也不靠写作谋生。他曾是一名才华横溢的学生，渴望尽可能多地学习，但对自己所学的东西却做出了严厉的评判：“我一完成学业，通常就会被纳入受过良好教育的人的行列……我发现自己陷入了如此多的疑惑和错误之中，在我看来，我本来打算学有所成，但我并没有从学习中获得任何好处。”

This isn’t Descartes’s message, and his life doesn’t correspond to any modern stereotype of intellectual life. He never held an academic position, and he didn’t make a living writing. He had been a brilliant student, eager to learn as much as he could, but passed a harsh judgment on what he had been taught: “As soon as I had finished my course of study, at which time it is usual to be admitted to the ranks of the well-educated . . . I found myself bogged down in so many doubts and errors, that it seemed to me that having set out to become learned, I had derived no benefit from my studies.”

笛卡尔决定放弃学术上的“推测”，直接从“世界巨著”中学习，去任何可能的地方：“我用我剩余的青春去旅行，访问宫廷和军队，与不同性格和地位的人交往，积累不同的经验，在偶然遇到的情境中考验自己。”

Descartes decided to turn away from scholarly “speculations” and learn directly from “the great book of the world,” going wherever that might take him: “I spent the rest of my youth travelling, visiting courts and armies, mixing with people of different character and rank, accumulating different experiences, putting myself to the test in situations in which I found myself by chance.”

武术

A Martial Art

笛卡尔非常明确地表达了他一生中最大的热情：“追求真理”。

Descartes was pretty explicit about his great passion in life: “seeking truth.”

我们很容易嘲笑、冷笑，从后现代的高度居高临下地看待它，假装这个观念真理的界限已经成为过去。然而，这也正是笛卡尔最值得我们学习的地方。

It’s easy to make fun of that, to sneer, to look condescendingly from the top of our postmodern heights, to pretend as if the notion of truth were a thing of the past. However, this is precisely where Descartes has the most to teach us.

只要我们把理性和逻辑混为一谈，只要我们将真理简化为社会和语言维度，只要我们只将其视为共识或权威的问题，我们就完全错过了笛卡尔方法的要点。

As long as we conflate rationality and logic, as long as we reduce truth to its social and linguistic dimensions, as long as we see it only as a matter of consensus or authority, we completely miss the point of the Cartesian approach.

对于笛卡尔来说，真理关乎生死。他完美地体现了数学心理学这一独特而强大的方面：他与真理的关系是物质的，几乎是肉体的：

For Descartes, truth was a matter of life and death. He perfectly embodies this singular and powerful aspect of mathematical psychology: his relationship to the truth is physical, almost carnal:

我始终渴望学会辨别真假，看清自己的行为，并充满信心地度过一生。

I constantly felt a burning desire to learn to distinguish the true from the false, to see my actions for what they were, and to proceed with confidence through life.

笛卡尔不太在意那些简单的真理，那些因为传统或某某人说是真理，或者仅仅因为它们看起来是真理，就被认为是真理的真理。他感兴趣的是那些坚实的真理，那些不会在一夜之间改变的真理，那些你可以依靠它变得更强大、更自信，在生活中做出正确选择的真理。

Descartes cared less for easy truths, those that are supposed to be true because tradition or such-and-such a person says so, or simply because they seem true. What interested him were solid truths, those that weren’t going to change overnight, the ones you can rely on to become stronger and more confident, to make the right choices in life.

他将真理视为一门武术，一种你培养起来并体现在行动中的本能。其他一切——哲学论证、没有利益的知识分子的“观点”——都只是空谈，对他毫无兴趣：“因为在我看来，我能从我们对影响我们的事情所做的推理中发现更多的真理，如果我们判断错误，这些事情很快就会给我们带来伤害，而不是从学者在书房里进行的对他毫无影响的推测中发现更多的真理。”

He approached truth as a martial art, an instinct you develop and that becomes embodied in action. Everything else—the philosophical arguments, the “opinions” of intellectuals with no skin in the game—was all just talk and of no interest to him: “For it seemed to me that I could discover much more truth from the reasoning that we all make about things that affect us and that will soon cause us harm if we misjudge them, than from the speculations in which a scholar engages in the privacy of his study, that have no consequence for him.”

从这个角度来看，笛卡尔对击剑的异常痴迷并不令人惊讶。20岁时，他克服了自认的“不喜欢击剑”的心理。他对“写书之道”有着深刻的理解，并就此撰写了两篇论文。手稿现已遗失，但一份尚存的摘要向我们展示了他对身体主宰问题的早熟兴趣。

In this light, Descartes’s uncommon obsession with fencing isn’t that surprising. At age twenty, he overcame his self-confessed “dislike [of] the business of writing books” and wrote a two-part treatise on the matter. The manuscript has since been lost, but a surviving summary shows us his precocious interest in the problem of mastery of the body.

如果《击剑艺术》今天出版，它可能会立即成为畅销书，至少如果这篇论文的第二部分能够实现它的承诺：“如果这是一场‘两个体型相同、力量相同、使用相同武器的人’之间的比赛，你如何才能始终击败你的对手。”

If The Art of Fencing were published today, it might become an instant best seller, at least if the second part of the treatise lived up to its promise: “how you can always beat your opponent” if it’s a competition of “two people of the same size, same strength, and same weapons.”

但在这里，我们现代的观点可能会让我们完全错过重点。对笛卡尔来说，击剑并不是好心人在周末在俱乐部里进行的一项消遣。它也不是智力竞赛的隐喻。击剑实际上是一门武术，一门战争艺术。

But here again, our modern perspective might make us completely miss the point. For Descartes, fencing wasn’t a hobby done on weekends at a club between well-meaning people. Nor was it a metaphor for intellectual jousting. Fencing was quite literally a martial art, an art of war.

二十二岁时，对自己的身体和方法充满信心，他开始从事一项被普遍认为对心智发展用处不大的职业：他报名当了一名雇佣兵。

At age twenty-two, confident in his body and his method, he started off in a career generally thought to be of little use in the development of the mind: he signed up as a mercenary.

理性的梦想

The Rational Dream

在笛卡尔的时代，欧洲的伟大思想家们被一个深刻的问题所困扰：太阳围绕地球旋转，还是地球围绕太阳旋转？

During Descartes’s time, the great minds of Europe were tormented by a deep question: does the Sun turn around the Earth, or is it the other way around?

要想想象“折磨”在这里到底意味着什么，需要付出真正的想象力。在当今世界，没有任何科学辩论能以同样的强度吸引人们的思想。也许有些人会争论地球是否平坦，但说科学因这个问题而“折磨”则过于夸张了。

It takes a real effort of the imagination to picture what “torment” might actually mean here. In today’s world, there’s no scientific debate that engages people’s minds with the same intensity. There might be some people who debate whether the Earth is flat, but it would be greatly exaggerated to say that science is “tormented” by the question.

哥白尼挑战了将地球置于宇宙中心的传统观点，他所引发的不仅仅是一场科学争论。他迫使基督教世界思考这个存在主义问题：真理一定是书本上写的吗？还是我们人类有能力自己发现真理？

By challenging the traditional view that placed the Earth at the center of the universe, Copernicus had sparked much more than a scientific quarrel. He had forced the Christian world to consider this existential question: is the truth necessarily what is written in books, or do we, as human beings, have the ability to discover it ourselves?

1619年11月10日至11日夜间，时年23岁的笛卡尔驻扎在德国多瑙河畔诺伊堡时，做了三个梦。

On the night of November 10–11, 1619, at the age of twenty-three, while he was stationed at Neuburg an der Donau in Germany, Descartes had a series of three dreams.

第一个相当复杂，特别是画中有一个甜瓜，有人想送给他，根据笛卡尔的说法，它代表着“孤独的魅力”。

The first was rather complicated and notably featured a melon someone wanted to give him and which, according to Descartes, represented “the charms of solitude.”

在第二个梦里，他感觉自己被雷击中。他惊醒过来，看到周围都是火花，好像房间着火了。在这里他又做出了解释：是“真理之灵”来占有他了。

In the second dream, he had the feeling of being struck by lightning. He woke up with a start and saw sparks all around him, as if the room were on fire. Here again he gave an interpretation: it was “the Spirit of Truth” that had come to take possession of him.

第三个梦就是所谓的清醒梦：在梦中，笛卡尔意识到自己在做梦，并在梦发生的同时开始解读自己的梦。

The third dream was what is called a lucid dream: in the middle of it, Descartes became aware that he was dreaming, and began to interpret his own dream while it was happening.

桌上出现了一本字典。他很高兴，并告诉自己这本字典可能会派上用场。但第二本书吸引了他的注意力，这是一本大型诗集，他正在翻阅，这时一个陌生人过来给他看了一首诗。笛卡尔认出了开头他读了拉丁诗人奥索尼乌斯的《毕达哥拉斯的是与否》一书后，开始在这本诗集中寻找它。

A dictionary appeared on the table. He was glad and told himself it might come in handy. But a second book drew his attention, a large collection of poetry that he was leafing through when a stranger came and showed him a poem. Descartes recognized the beginning of “The Pythagorean Yes and No” by the Latin poet Ausonius, and he began to look for it in the collection.

过了一会儿，笛卡尔发现字典被损坏了。然后，这个人和书都消失了。笛卡尔没有醒来，他把字典解释为科学的象征，把诗集解释为哲学和智慧的象征。梦的内容正是如此：需要在受到诗人技巧的启发的同时重建科学，诗人通过“热情的神圣性”和“想象力的力量”，能够发掘“智慧的种子（存在于所有人的精神中，就像石头中的火花一样）”。

A bit later, Descartes realized that the dictionary was damaged. Then the man and the books disappeared. Without waking up, Descartes interpreted the dictionary as a symbol of science and the poetry collection as a symbol of philosophy and wisdom. The substance of the dream was precisely that: it is necessary to reconstruct science while being inspired by the techniques of poets who, through “the divinity of enthusiasm” and “the force of imagination,” are able to uncover “the seeds of wisdom (which are found in the spirit of all men, like the sparks of fire in stone).”

笛卡尔就这样发明了理性。他醒来后确信真理之灵降临到他身上，要“打开所有科学的宝藏”，揭示真理不在书本中，而是在我们的头脑中。我们有能力通过我们的思想力量自己发现它。

Descartes thus invented rationality. Upon awakening, he was convinced that the Spirit of Truth had descended upon him to “open the treasures of all the sciences” by revealing that truth is not to be found in books, but in our heads. We have the ability to discover it ourselves, by the power of our thought.

对于笛卡尔来说，证据是真理的核心标准。证据不是表面证据，即往往是错误的初始直觉，而是通过刻意和系统的澄清、言语化和解释的努力构建的证据，其目的是使事物完全可理解，直到它们变得完全透明：“我们非常清楚和明确地设想的事物总是真实的。”

For Descartes, evidence is the core criterion of truth. Not superficial evidence, the initial intuition that is often false, but evidence constructed through a deliberate and systematic effort of clarification, verbalization, and explanation, which aims to make things perfectly intelligible until they become fully transparent: “Things that we conceive of very clearly and distinctly are always true.”

真正的数学

The True Mathematics

这一启示指导了他的余生。次年，即 1620 年，笛卡尔放弃了军事生涯，全身心投入科学研究。他从算术和几何开始，这是“所有科学中最简单、最清晰的学科”，他以一流的成绩取得了辉煌的开端。

This revelation would guide the rest of his life. The following year, in 1620, Descartes abandoned his military career to dedicate himself to science. He began with arithmetic and geometry, “the easiest and clearest of all the sciences,” where he made an auspicious debut with first-class results.

他认为数学是一切知识的基础，我们已经习惯的技术意义（自十七世纪以来，数学形式主义已经成为科学的基本工具之一），但同时，最重要的是，它以一种基本的、原始的意义存在于人类心理的深处。

He saw in mathematics the basis of all knowledge, not only in the technical sense we’ve grown accustomed to (since the seventeenth century, mathematical formalism has become one of the basic tools of science), but also, and above all, in a basic, primordial sense, seated in the depths of human psychology.

对于笛卡尔来说，理解数学的经验是理解“理解”真正含义的唯一方法。它具有一种强度和令人困惑的独特性，就像启示一样对我们起作用。数学是一种精神觉醒。它教会我们识别正确的身体感觉，这将引导我们走上知识之路。除非我们亲自遇到这种结晶和半透明的真理形式，否则不可能知道事物“清晰明了”的含义，不可能理解笛卡尔试图告诉我们什么，而且，根据他的说法，不可能进入真正的知识途径。

For Descartes, the experience of understanding mathematics is the sole means of understanding what “understanding” really means. It has an intensity and bewildering uniqueness that acts on us like a revelation. Mathematics is a spiritual awakening. It teaches us to recognize the correct physical sensation, that which will guide us on the path to knowledge. And until we have personally encountered this crystalline and translucid form of truth, it is impossible to know what it means for a thing to be “clear and distinct,” it is impossible to understand what Descartes is trying to tell us, and, according to him, it is impossible to enter into a real approach to knowledge.

1628 年，他开始撰写《心智指导规则》，这是他首次尝试阐述自己的方法。笛卡尔从未选择出版的这本书预示了近十年后他撰写的《方法论》 。

Toward 1628 he began writing Rules for the Direction of the Mind, his first attempt to lay out his method. This text, which Descartes never chose to publish, prefigures Discourse on Method, which he would write nearly ten years later.

在书中，他将官方数学描述为一件“外衣”，必须先脱掉它才能了解其真正内容：“如果这些规则只能用来解决算术家和几何学家用来消磨时间的那些毫无意义的问题，我就不会如此看重它们，因为在这种情况下，我所能取得的成就就是比他们更精妙地涉猎琐事。”

In it, he presents official mathematics as an “outer garment” that must be removed before its real substance can be accessed: “I would not value these Rules so highly if they were good only for solving those pointless problems with which arithmeticians and geometers are inclined to while away their time, for in that case all I could credit myself with achieving would be to dabble in trifles with greater subtlety than they.”

笛卡尔将他所谓的“真正的数学”与教科书中的“幼稚和无意义”的东西进行了对比。在学习算术和几何的过程中，笛卡尔对这种脱节感到沮丧：“在这两个学科上，我都没有遇到能完全让我满意的作家。我读了很多关于数字的书，一旦我自己计算过，就会发现它们是正确的。作家们“他们把许多几何真理在我眼前演绎出来，并通过逻辑论证推导出来。但他们似乎并没有让我充分明白为什么这些事情会这样，以及它们是如何被发现的。”

Descartes contrasts what he calls the “true mathematics” with the “childish and pointless” stuff you find in textbooks. In his own effort to learn arithmetic and geometry, Descartes had grown frustrated at the disconnect: “In neither subject did I come across writers who fully satisfied me. I read much about numbers which I found to be true once I had gone over the calculations for myself. The writers displayed many geometrical truths before my very eyes, as it were, and derived them by means of logical arguments. But they did not seem to make it sufficiently clear to my mind why these things should be so and how they were discovered.”

古希腊哲学家赋予数学特殊的地位。他们把数学作为所有哲学和科学的先决条件。传说柏拉图学院的入口处刻有以下这句话：“不懂几何学的人不得入内。”

The ancient Greek philosophers granted a special place to mathematics. They made it the prerequisite of all philosophy and science. Legend has it that the following phrase was engraved at the entry to Plato’s Academy: “Let no one ignorant of geometry enter.”

对于笛卡尔来说，如果古希腊人只知道“幼稚而无意义”的东西，那么这根本说不通。他们一定熟悉“真正的数学”：“我开始怀疑他们熟悉的是一种与今天流行的数学截然不同的数学。”

For Descartes, this wouldn’t have made any sense if the ancient Greeks had only known about the “childish and pointless” stuff. They must necessarily have been acquainted with the “true mathematics”: “I came to suspect that they were familiar with a kind of mathematics quite different from the one which prevails today.”

笛卡尔甚至解释了为什么这种特殊的数学没有传给我们。据他说，古希腊人故意选择保密，因为它太容易、太简单了，泄露它会损害他们的知识威望：

Descartes even has an explanation as to why this special kind of math wasn’t passed on to us. According to him, the ancient Greeks intentionally chose to keep it secret, because it was too easy and too simple, and disclosing it would have damaged their intellectual prestige:

我开始认为，这些作家自己后来以一种有害的狡猾手段隐瞒了这些数学知识，众所周知，许多发明家在涉及他们自己的发现时都这样做。他们可能担心，正因为他们的方法如此简单易行，如果泄露出去，他们就会贬值；所以为了赢得我们的钦佩，他们可能向我们展示了一些用巧妙的论证证明的毫无意义的真理，作为他们方法的成果，而不是教给我们方法本身，因为这可能会打消我们的钦佩之情。

I have come to think that these writers themselves, with a kind of pernicious cunning, later suppressed this mathematics as, notoriously, many inventors are known to have done where their own discoveries were concerned. They may have feared that their method, just because it was so easy and simple, would be depreciated if it were divulged; so to gain our admiration, they may have shown us, as the fruits of their method, some barren truths proved by clever arguments, instead of teaching us the method itself, which might have dispelled our admiration.

重新与我们内心的真相联系起来

Reconnecting with the Truth within Us

《心灵指引规则》是一本富有远见的著作，它预见了我们从本书一开始就讨论的主题。笛卡尔甚至阐述了这个深刻的真理，这在当时看来是如此现代ern，阻止我们前进的主要障碍是心理上的绊脚石。

Rules for the Direction of the Mind is a visionary text that anticipates the themes that we’ve addressed from the outset of this book. Descartes even formulated this profound truth, which seems so modern, that the principal objects stopping us are psychological stumbling blocks.

他的方法非常深邃。为了找回这种古老知识的道路，他邀请我们重新与原始的清醒联系起来，即“自然植入人类头脑中的真理的原始种子，它们在那个单纯而天真的时代蓬勃发展——而这些种子在我们不断阅读和听到各种错误时被扼杀了。”

His approach is deeply meditative. To recover the path of this ancient knowledge, he invites us to reconnect with our primitive lucidity, the “primary seeds of truth naturally implanted in human minds [that] thrived vigorously in that unsophisticated and innocent age—seeds which have been stifled in us through our constantly reading and hearing all sorts of errors.”

他指出，困难是情感和非智力层面的。困难源于我们的社会需求，即让别人和我们自己相信我们理解了我们实际上并不明白的事情。这有点像背越式跳高：要采取正确的姿势，你必须克服让我们相信自己处于危险中的逃避反射。我们必须摆脱错误的理解和虚张声势的倾向。

He notes that the difficulty is of an emotional and nonintellectual order. It arises from our social need to make believe, for others as for ourselves, that we understand what in reality we do not. It’s a bit like the Fosbury flop: to adopt the correct position, you have to overcome the flight reflex that makes us believe that we’re putting ourselves in danger. We have to rid ourselves of our false understanding and our tendencies to bluff.

这种掩饰的本能尤其影响知识分子：“因为他们认为，承认自己对任何事情都无知，对一个有学问的人来说，都是不合适的，所以他们已经习惯于详细阐述他们人为的学说，以至于他们逐渐相信了这些学说，并把它们当作真理。”

This instinct for dissimulation particularly affects intellectuals: “Because they have thought it unbecoming for a man of learning to admit to being ignorant on any matter, they have got so used to elaborating their contrived doctrines that they have gradually come to believe them and to pass them off as true.”

我们如此缺乏安全感，以至于我们甚至放弃了真正理解的可能。因为我们拒绝相信它真的可以很简单，所以我们在相反的方向、在复杂和困难中寻找知识。

Our insecurity is such that we’ve abandoned the idea that real understanding is even possible. Because we refuse to believe that it could really be simple, we look for knowledge in the opposite direction, in the complicated and difficult.

笛卡尔意识到他的建议与我们的本能相悖，因此他小心翼翼地强调了这一信息：

Descartes realized how much his recommendations went against our instincts, so he took care to hammer home the message:

如果没有直觉或推理，我们就不可能获得知识。

We can have no knowledge without mental intuition or deduction.

整个方法完全在于对对象进行排序和排列，如果我们要发现一些真理，就必须集中我们的心眼。

The whole method consists entirely in the ordering and arranging of the objects on which we must concentrate our mind’s eye if we are to discover some truth.

如果我们首先将复杂而晦涩的命题逐步简化为更简单的命题，然后从最简单的命题的直觉开始，尝试通过相同的步骤逐步掌握所有其他命题，那么我们就完全遵循了这种方法。

We shall be following this method exactly if we first reduce complicated and obscure propositions step by step to simpler ones, and then, starting with the intuition of the simplest ones of all, try to ascend through the same steps to a knowledge of all the rest.

但许多人要么不去思考这条规则规定的内容，要么完全忽视它，或者认为他们不需要它……就好像他们试图一下子从一栋建筑的底部爬到顶部，却无视或没有注意到为此目的而设计的楼梯。

But many people either do not reflect upon what the Rule prescribes, or ignore it altogether, or presume that they have no need of it . . . as if they were trying to get from the bottom to the top of a building at one bound, spurning or failing to notice the stairs designed for that purpose.

如果在一系列需要检查的事物中，我们遇到了我们的智力无法充分直观地理解的事物，我们就必须就此停下来。

If in the series of things to be examined we come across something which our intellect is unable to intuit sufficiently well, we must stop at that point.

“我的坦诚将受到大家的赞赏”

“My candour will be appreciated by everyone”

然而，规则没有明确指出的是，笛卡尔面临着一个重大问题。这个问题是他的哲学的核心，但他却从未能够解决它：他自己尝试过这种方法，他知道它有效，但他无法解释为什么。

What the Rules didn’t make explicit, however, was that Descartes was confronted with a major problem. This problem is central to his philosophy, and yet he was never able to resolve it: he had tried this method himself, he knew that it worked, but he could never explain why.

这是所有数学家都面临的问题，我们已经讨论过这个问题。我们很难谈论我们在头脑中做出的看不见的动作。很难用别人能理解的具体术语描述这种方法。很难找到一个合理的解释来解释它有效。我们用来分享心理体验的词汇量太少了，以至于我们很快就会给人留下什么都说的印象，只不过是障眼法。

It’s a problem that all mathematicians face, and one we’ve already discussed. It’s difficult to talk about the unseen actions that we make in our heads. It’s difficult to describe the method in tangible terms that others will make sense of. It’s difficult to find a rational explanation for the fact that it works. The vocabulary that we possess to share our mental experiences is so poor that we rather quickly give the impression of saying whatever, nothing but smoke and mirrors.

老实说，当笛卡尔说“真理之灵”降临并占据了他，或者“真理的原始种子自然地植入人类心灵”时，很难认真对待他的话。然而，这是他能够给出的唯一解释。这这使他采取了二元论的立场：他认为心灵和身体是分离的。我们的心灵是非物质的，是上帝按照他的形象创造的，因此能够像变魔术一样获得真理。

And honestly, it’s hard to take Descartes seriously when he speaks of the “Spirit of Truth” descending to take possession of him, or the “primary seeds of truth naturally implanted in human minds.” It is, however, the only explanation that he was in a position to give. This lead him to adopt a dualist stance: he imagined a separation between mind and body. Our mind is of an immaterial nature, created by God in his image, and thus capable of attaining Truth as if by magic.

你可以对这个信念想什么就想什么，但你不能把它视为理所当然。无论如何，它并没有达到笛卡尔想要强加给自己的那种严格程度：只接受那些显而易见、不容置疑的真理。

You can think what you like of this belief, but you can’t take it for granted. At any rate, it’s not at the level of rigor that Descartes aspired to impose upon himself: to accept as true only that which is so evident that it’s impossible to doubt.

从远处就可以看出问题的严重性。在 17 世纪，不可能基于神经可塑性做出解释。在笛卡尔时代，人们对​​人体的了解相当有限。他本人认为心脏是加热血液的熔炉。当时，医疗实践仍以狂欢节的名义进行，然后才进行灌肠和放血。

From a distance, you can see the extent of the problem. An explanation based on neuroplasticity was impossible to formulate in the seventeenth century. In Descartes’s time, knowledge of the human body was rather limited. He himself thought of the heart as a furnace that served to heat the blood. At the time, medical practice still dressed itself in the guise of a carnival show before proceeding with enemas and bloodletting.

在历史上的伟大数学家中，笛卡尔远不是唯一一个对创造力给出超自然解释的人。在《收获与播种》之后，格罗滕迪克写了一本神秘的著作《梦的钥匙》 ，书中说上帝来到他的内心做梦，向他展示真理。

Among the great mathematicians in history, Descartes is far from being the only one to give a supernatural explanation for creativity. In the Key to Dreams, a mystic text that he wrote after Harvests and Sowings, Grothendieck says that God came to dream inside of him to show him the truth.

斯里尼瓦瑟·拉马努金（我们将在书的结尾详细讨论他）说，他的定理是由他的家族女神娜玛吉里·泰亚尔（Namagiri Thayar）向他揭示的，娜玛吉里·泰亚尔是他梦中反复出现的角色。

Srinivasa Ramanujan, whom we’ll talk more of at the end of the book, said that his theorems were revealed to him by his family goddess Namagiri Thayar, a recurring character in his dreams.

我承认这些解释总是让我有点怀疑。我们将在最后几章中回顾它们。

I admit that these explanations have always left me a bit skeptical. We’ll come back to them in the final chapters.

也许是因为他怀疑自己很难说服别人，所以笛卡尔放弃了《心灵指导规则》。

It’s perhaps because he suspected he’d have a hard time convincing others that Descartes abandoned Rules for the Direction of the Mind.

他没有直接阐述自己的方法，而是选择将其付诸实践。他从事数学、物理学和生物学研究。他游历欧洲。他在阿姆斯特丹的屠夫区呆了好几年，这让他有机会接触动物尸体进行解剖。解剖学是他的主要兴趣之一。

Rather than directly stating his method, he chose to put it in practice. He pursued research in mathematics, physics, and biology. He traveled across Europe. He spent a number of years in Amsterdam, in the butchers’ district, which gave him access to animal cadavers for dissection. Anatomy was one of his primary interests.

到 17 世纪 30 年代初，笛卡尔已准备好撰写一本雄心勃勃的《世界论》，旨在解释所有自然现象。

By the early 1630s, Descartes was readying an ambitious Treatise on the World that was to explain all natural phenomena.

但发生的事情让整个计划受到了质疑。1633 年 2 月，伽利略因捍卫哥白尼理论而被判异端罪并被软禁。

But something happened that called the whole plan into question. In February 1633, Galileo was convicted of heresy and placed under house arrest for having defended Copernicus’s theories.

笛卡尔从不拿个人安全开玩笑，他停止了研究和论文的发表。这种对安全的担忧已经导致他自我流放至荷兰，荷兰是一个新教国家，因此不受宗教裁判所的管辖。

Descartes, who never joked about his personal security, halted his research and the publication of his treatise. This same preoccupation with security had already led to his self-exile in the Netherlands, a Protestant nation and thus outside the reach of the Inquisition.

他决定只发表部分内容，而不是全部内容，因为全部内容几乎可以保证定罪。他从论文中摘录了最不可能引起问题的章节。

He resolved to pursue a partial publication, rather than a complete one, which would almost have guaranteed a conviction. He extracted from his treatise those chapters least likely to cause problems.

《论正确进行理性和在科学中寻求真理的方法》是笛卡尔 1637 年匿名出版的文集的序言。这篇序言之后的三篇短篇论文（《几何学》（我们已经提到过的数学论文）、《光学》（其标题不言自明）和《流星》（试图解释风、闪电和彩虹等现象）证明了他的方法的有效性。

A Discourse on the Method of Correctly Conducting One’s Reason and of Seeking Truth in the Sciences is Descartes’s introduction to the collection he published, anonymously, in 1637. The three short treatises that followed this introduction (Geometry, the mathematical treatise we already mentioned, Optics, whose title is self-explanatory, and Meteors, which sought to explain phenomena such as wind, lightning, and rainbows) are presented as proof of the effectiveness of his method.

与以极其专横、权威的风格撰写的《心智指导规则》相反， 《方法论》的开头却出奇地谦逊。

As opposed to Rules for the Direction of the Mind, written in an extremely preemptory and authoritative style, Discourse on Method begins in a strikingly modest fashion.

笛卡尔似乎意识到他的故事有点夸张，人们可能难以接受。他自己似乎并不完全相信。从书的开篇开始，他就承认其中存在一些矛盾。一方面，他很清楚自己的科学工作至关重要。在这方面，他并不虚伪谦虚。另一方面，他觉得自己并不是特别有天赋。

Descartes seemed to have realized that his story was a bit much and that people might have a hard time swallowing it. He didn’t seem to completely believe it himself. From the opening pages of his book, he confessed that there was something of a contradiction. On the one hand, he was well aware of the utmost importance of his scientific work. In that respect, he didn’t suffer from false modesty. On the other hand, it seemed to him that he wasn’t particularly gifted.

他得出结论，他的成功只能解释他用自己偶然发现的方法。但他意识到，这似乎好得令人难以置信：

He drew the conclusion that his success could only be explained by the method he had come across by chance. He recognized, however, that this might seem too good to be true:

然而，我也可能错了，我可能把铜片和玻璃片误认为金子和钻石。我知道我们自己犯错的可能性有多大……所以我的目的不是教每个人必须遵循的正确推理方法，而只是展示我尝试过的方式。

It is, however, possible that I am wrong, and that I am mistaking bits of copper and glass for gold and diamonds. I know how likely we are to be wrong on our own account. . . . So my aim here is not to teach the method that everyone must follow for the right conduct of his reason, but only to show in what way I have tried to conduct mine.

简而言之，他不是来给我们上课的。和《收获与播种》一样，《方法论》是一本自传，一份我们不应该只看表面的证明：“我希望它对某些人有用，但不会伤害任何人，我的坦诚会得到所有人的赞赏。”

In short, he wasn’t there to give us lessons. Like Harvests and Sowings, Discourse on Method is an autobiography, a testimonial that we shouldn’t necessarily take at face value: “I hope that it may prove useful to some people without being harmful to any, and that my candour will be appreciated by everyone.”

内心的怀疑

Visceral Doubt

笛卡尔的怀疑有点像本·安德伍德的点击：人们很难相信它会起作用，以至于他们甚至懒得尝试，或者在它开始起作用之前就放弃了。

Descartes’s doubt is a bit like Ben Underwood’s clicking: people find it so hard to believe that it works that they don’t even bother trying, or they give up before it’s started to work.

在数学界之外，我相信我从未遇到过真正认真对待怀疑的人。

Outside of the mathematical community, I don’t believe I’ve ever come across anyone who has really taken doubt seriously.

真是浪费！一位伟大的数学家不辞辛劳地告诉我们他是如何做到的，尽管他认为自己智力平平。他白纸黑字地表明，他的论述没有理论基础，但应该被视为“值得效仿的例子”的来源。一代又一代的学生被打得头破血流，被迫将《论语》视为一篇哲学论文。几乎没有人愿意真正尝试一下。

What a waste! A great mathematician takes the trouble to tell us how he got there despite being what he deemed as of average intelligence. He set down in black and white that his account has no theoretical basis but should be taken as a source of “examples worthy of imitation.” Generations of students have been beaten over the head and forced to regard the Discourse as a philosophical treatise. And almost no one has taken the trouble to try it for real.

在结束本章之前，让我们花点时间澄清一下笛卡尔怀疑到底是什么，以及每个人可以从中获得的个人利益毕竟，这是对另一个关键概念（到目前为止我们才刚刚开始讨论）的最佳介绍：数学证明的概念。

Before ending this chapter, let’s take the time to clarify what Cartesian doubt really is, and the personal benefits everyone can get from it. After all, this is the best possible introduction to another key concept that up to now we’ve only begun to address: the concept of the mathematical proof.

在学校里，我们被教导说笛卡尔怀疑是一种有条不紊的怀疑。这是一种说法，说这种方法是基于怀疑的。但这种表达方式可能会导致混淆。人们很容易误解，认为你必须以一种有条不紊的方式去怀疑。这是完全不可能的，原因和你不可能以一种有条不紊的方式坠入爱河是一样的。你只能凭直觉去怀疑。所有笛卡尔怀疑都是本能的。

At school we’re taught that Cartesian doubt is a methodical doubt. It’s a way of saying that the method is based on doubt. But this expression can lead to confusion. It’s easy to take it the wrong way, thinking that you have to doubt in a methodical fashion. That’s completely impossible, for the same reason that it’s impossible to fall in love in a methodical fashion. You can only doubt with your gut. All Cartesian doubt is visceral.

想要有条不紊地进行怀疑就会混淆系统 2（机械思维）和系统 3（直觉和逻辑之间的对话）。

Wanting to doubt in a methodical fashion is to confuse System 2, mechanical thinking, with System 3, the dialogue between intuition and logic.

笛卡尔并不反对系统 2。例如，他建议列清单，这样你就不会忘记任何事情。但怀疑的过程不属于系统 2。你不能用言语来怀疑，你只能默默地、在脑海里怀疑。怀疑是个人的、私密的。如果你只是假装怀疑，如果你不全力以赴，如果你不冒险，那就一文不值。

Descartes wasn’t against System 2. For example, he recommended making lists so that you don’t forget anything. But the process of doubt doesn’t belong to System 2. You can’t doubt with words, you can doubt only silently, in your head. Doubt is personal and intimate. If you only pretend to doubt, if you don’t go all the way, if you don’t take the plunge, it’s worth nothing.

在发明怀疑论时，笛卡尔将自己置于官方知识的对立面。他生活在一个真理仍然与权威混为一谈的世界：真理是传统，是书本上所写的东西。科学仍然是亚里士多德方法的继承者，已有两千多年的历史。这种方法包括汇编和尝试构建一堆据称 99% 真实的东西（或者某个所谓严肃的人说这些东西 99% 真实，或者 80% 真实，或者 51% 真实——你永远不知道它们到底是真是假）。

In inventing doubt, Descartes positioned himself in opposition to official knowledge. He lived in a world where truth was still conflated with authority: truth was tradition, what was written in books. The sciences were still the heirs of the Aristotelean approach, over two thousand years old. That approach consisted of compiling and trying to structure a hodgepodge of things supposed to be 99 percent true (or that someone supposedly serious had said were 99 percent true, or 80 percent, or 51 percent—you never really knew).

例如，当亚里士多德解释地球为什么是圆的时候，他收集了一堆从其他来源获取的混合论据。论据越多，就越有说服力，直到他悄悄地解释说，非洲有大象，亚洲有大象，因此两端相接，所以地球是圆的。

For example, when Aristotle explained why the Earth was round, he amassed a bunch of mixed arguments taken from other sources. The more arguments there were, the more convincing it was supposed to be, up to the point where he quietly explains that there are elephants in Africa and elephants in Asia, therefore the two ends meet, and therefore the Earth is round.

怀疑就是对一个论点进行嗅探，并感觉到有什么不对劲。它让你问自己，“什么？真的吗？”

To doubt is to give an argument the sniff test and sense that there’s something off. It’s allowing yourself to ask, “What? Really?”

笛卡尔的立场其实很简单：他认为，可能性很大的事情，即使 99.99% 确定但不是 100%，也可能很有趣，但从科学的角度来看，它毫无价值，因为你无法以此为基础。（正如我们将看到的，这是一种极端的观点，在最纯粹的形式下，它存在很大问题。现代科学实际上是建立在很可能但不是 100% 确定的事情之上的，有充分的理由以这种方式进行。）

Descartes’s position is really quite simple: he states that what is probable, even 99.99 percent sure but not 100 percent, might be interesting, but from a scientific perspective it’s worth nothing, because you can’t build on it. (As we’ll see, this is an extreme view that, in its purest form, is highly problematic. Modern science is in fact built on things that are very likely yet not 100 percent certain, and there are good reasons to proceed this way.)

教授笛卡尔怀疑论很难，因为它既不是知识，也不是论证方式，因此无法评估。没有人能在一张纸上怀疑。怀疑是一种秘密的运动活动，一种看不见的行为。怀疑某件事就是能够想象出一种情景，即使看似不可能，但这件事可能不真实。

Teaching Cartesian doubt is difficult because it’s neither a knowledge nor a mode of argumentation, and is therefore impossible to evaluate. No one can doubt on a piece of paper. Doubt is a secret motor activity, an unseen action. To doubt something is to be able to imagine a scenario, even seemingly improbable, where the thing could be untrue.

笛卡尔要求我们不仅要怀疑别人说的话，而且最重要的是要怀疑我们自己的确定性。这是他的方法的核心，也是我们最难理解的地方。怀疑我们自己的确定性就像头朝下、仰面朝上地跳高：我们的本能告诉我们这很危险。我们害怕暴露我们身体上的弱点，害怕跌倒，害怕掉进一个无底深渊，在那里我们将不再对任何事情有信心。而且我们完全看不到我们能从中得到什么。

Descartes asks us to doubt not only what others say, but also, and above all, our own certitudes. This is at the heart of his method, and it’s where we have the hardest time following him. To place in doubt our own certitudes is like doing a high jump headfirst and on your back: our instinct tells us it’s dangerous. We’re afraid of exposing our physical vulnerabilities, of foundering, of falling into a bottomless pit where we’ll no longer have confidence in anything. And we absolutely don’t see what we stand to gain from it.

如果你从未真正体验过数学，你就会倾向于认为它是一个无底洞。笛卡尔所要求的确定性水平似乎是不可能达到的。

If you’ve never really experienced math, you’ll have a tendency to believe it’s a bottomless pit. The level of certainty demanded by Descartes seems impossible to attain.

数学为我们提供了可以绝对相信的真理的例子。它不仅包括表面的真理，如 2 + 2 = 4，还包括深刻的真理，这些真理极其有趣，而且一点也不明显。我们将在下一章中给出几个引人注目的例子。

Mathematics gives us examples of truths in which we can have absolute confidence. It’s not just about superficial truths, like 2 + 2 = 4, but also profound truths, truths that are extremely interesting and not at all immediately apparent. We’ll give several striking examples in the next chapter.

只有通过不懈地对抗怀疑，迫使你澄清和具体化每一个细节，直到一切都变得透明，你才能最终创造出显而易见的东西。怀疑是一种精神澄清的技巧。它的作用是建设而不是破坏。

It’s only through a relentless confrontation with doubt that forces you to clarify and specify each detail until it all becomes transparent that you’re finally able to create obviousness. Doubt is a technique of mental clarification. It serves to construct rather than destroy.

“一种排除一切恐惧的好奇态度”

“An attitude of curiosity that excludes all fear”

在数学之外，由于与语言和大脑功能相关的深刻原因，笛卡尔所要求的确定性水平似乎是不可能达到的，而这些原因直到最近才被理解（我们稍后会回到这个问题）。

Outside of mathematics, the level of certainty required by Descartes turns out to be impossible to attain for profound reasons that relate to how language and our brains function, and that have only recently been understood (we’ll come back to this later).

这丝毫不影响笛卡尔怀疑论的力量和效力。 《方法论》中关于个人发展的讯息远远超出了数学领域和对永恒真理的探索。

That takes nothing away from the strength and potency of Cartesian doubt. The message of personal development in Discourse on Method goes well beyond the field of mathematics and the search for eternal truths.

这句话只有在笛卡尔的性格和动机下才有意义。他讨厌“那些为了怀疑而怀疑，假装永远无法做出决定的怀疑论者”。正如我们所见，他的“强烈愿望”恰恰相反：“满怀信心地度过一生。”

This message makes sense only in light of Descartes’s personality and his motivations. He hated “those sceptics who doubt for doubting’s sake, and pretend to be always unable to reach a decision.” His “burning desire,” as we’ve seen, was the exact opposite: “to proceed with confidence through life.”

他对怀疑的态度与他对直觉的品味密切相关，他将直觉定义为“一种清晰而专注的头脑的概念，它如此简单和明确，以至于不存在任何怀疑的余地。”

His approach to doubt is closely tied to his taste for intuition, which he defined as “the conception of a clear and attentive mind, which is so easy and distinct that there can be no room for doubt.”

因此，怀疑对应于直觉的阴影区。要真正怀疑某件事，你不能只是声称你怀疑，你必须真诚地相信这件事可能不是真的。为了做到这一点，你必须在头脑中构建一个图像，表明有怀疑的空间。如果你做不到这一点，你就不能怀疑——你是肯定的，就像你可以肯定 2 + 2 = 4 一样。但一旦你能够想象当事情可能不真实时，怀疑会立即启动重新配置你的心理表象的过程。

Doubt thus corresponds to the shadow zones of intuition. To really doubt something, you can’t just claim that you doubt, you have to sincerely believe that this thing might not be true. In order to do that, you have to construct an image in your head that shows there’s a place for doubt. If you can’t do that, you can’t doubt—you’re certain, like you can be with 2 + 2 = 4. But once you’re able to imagine a scenario where the thing can be untrue, doubt immediately starts the process of reconfiguring your mental representations.

由于笛卡尔版本的怀疑激发了想象力，它类似于前几章描述的技术，只不过它关注的不是数字或几何形式，而是真理本身。

Because it mobilizes the imagination, the Cartesian version of doubt resembles techniques described in the previous chapters, except that instead of being concerned with numbers or geometric forms, it’s concerned with truth itself.

笛卡尔怀疑是一种重新编程直觉的通用技术。

Cartesian doubt is a universal technique for reprogramming your intuition.

因此，在笛卡尔的著作中发现与格罗滕迪克和瑟斯顿给出的建议非常相似的建议并不奇怪。例如，他要求我们全身心投入到我们的认知发展中：“我们必须利用智力、想象力、感知和记忆所能提供的一切帮助，首先，直观地理解简单的命题。”

It’s therefore not surprising to find in Descartes’s writing very similar advice to that given by Grothendieck and Thurston. He asks us, for example, for total physical commitment in service of our cognitive development: “We must make use of all the aids which intellect, imagination, sense-perception, and memory afford in order, firstly, to intuit simple propositions distinctly.”

笛卡尔从未明确地谈论过心理可塑性。这一概念直到他去世几个世纪后才被概念化。但他对自己方法的好处的描述并没有留下太多的歧义：“我相信，在实践过程中，我的思想逐渐习惯于更清晰、更明确地构想其对象。”

Descartes never speaks explicitly of mental plasticity. This notion would be conceptualized only many centuries after his death. But his account of the benefits of his method doesn’t leave much room for ambiguity: “I believed that, in practising it, my mind was gradually getting used to conceiving of its objects more clearly and distinctly.”

笛卡尔发现，当我们真诚地尝试反省时，当我们关注自己的认知失调时，当我们强迫自己抓住最短暂的心理图像并用语言表达出来时，当我们有勇气面对想象中的内部矛盾时，当我们有足够的冷静和自制力超越偏见看清事物的本来面目时，我们的心理表象就会得到改变，变得更加强大、坚实、连贯和有效。

Descartes discovered that when we make a sincere attempt at introspection, when we’re attentive to our cognitive dissonance, when we force ourselves to grasp our most fleeting mental images and put words to them, when we have the courage to face the internal contradictions of our imagination, when we have enough calm and self-control to look beyond our prejudices and see things as they really are, it has the result of modifying our mental representations, of making them more powerful, solid, coherent, and effective.

笛卡尔发现的是人体的一个特性。

What Descartes discovered is a property of the human body.

笛卡尔的视觉词汇的卓越性在此例中令人震惊。当他说真理是“清晰而明确”的东西时，你几乎可以说他给它赋予了神经学定义。他的方法让人想起瑟斯顿和本·安德伍德的方法：这是一种学习观察的方法。

The preeminence of the vocabulary of vision in Descartes is in this instance striking. When he states that truth is that which is “clear and distinct,” you could almost say that he’s giving it a neurological definition. His method evokes that of Thurston and Ben Underwood: it’s a method for learning to see.

如果正确运用怀疑，怀疑可以引发一种深刻的理解状态，这种状态让笛卡尔感到惊讶，也让所有经历过怀疑的人感到惊讶。这种体验会让你焕然一新，而这本身就值得付出努力。

Practiced right, doubt can induce a state of profound comprehension that amazed Descartes, as it amazes all those who experience it. It’s an experience that leaves you transformed, and that in itself is well worth the effort.

怀疑不仅是笛卡尔成就背后的秘密，也是他令人难以置信的厚颜无耻的秘密。从这个角度来看，《方法论》是一堂自信大师课。他的理性版本是具体的、个人的，植根于我们最深切的愿望。整个目的是让我们变得更强大：“这种自信是心态的一面，而心态的另一面是对怀疑的开放：一种好奇的态度，它排除了对自己错误的所有恐惧，使我们能够发现并不断纠正错误。”

Doubt is not only the secret behind Descartes’s achievements, it’s also the secret of his incredible chutzpah. Seen in this way, Discourse on Method is a master class in self-confidence. His version of rationality is concrete, personal, rooted in our deepest aspirations. The whole point is to make us stronger: “This assurance is one side of a mindset, whose other side is an openness to doubt: an attitude of curiosity that excludes all fear as regards one’s own mistakes, that allows us to detect and constantly correct them.”

这最后一句引文出自格罗滕迪克的《收获和播种》，完美地概括了笛卡尔的这一基本教训，并抓住了数学精神的独特之处。

This last quote, taken from Grothendieck’s Harvests and Sowings, perfectly summarizes this fundamental lesson from Descartes, and captures a unique aspect of the mathematical ethos.

傲慢的人喜欢被反驳，爱炫耀的人在你证明他们错了的时候会微笑，教条主义者随时准备改变主意：我只在非常优秀的数学家中遇到过这种独特的态度。

Arrogant people who love being contradicted, show-offs who smile when you prove them wrong, dogmatists ready to change their mind in a heartbeat: I’ve encountered this singular attitude only among very good mathematicians.

15

敬畏与魔力

15

Awe and Magic

如果数学家的思维活动是可见的，那么研究机构就会有玻璃墙。路人会停下来观看，就像他们会看到正在风筝冲浪或攀岩的人一样。在高中，数学会比滑板更受欢迎。

If the mental actions of mathematicians were visible, research institutes would have glass walls. Passersby would stop to look, as they would at people who are kitesurfing or rock climbing. In high school, mathematics would be more popular than skateboarding.

失去了模仿的可能性，我们失去的不仅仅是我们的主要学习方法，我们还失去了我们欲望的主要驱动力。

In losing the possibility of imitation, we lose much more than our main learning method. We also lose our main driver of desire.

当你还是个孩子时，没人需要强迫你骑自行车。没人需要说服你骑自行车对你以后的生活大有裨益，或者骑自行车会让你的简历增色不少。

When you were a child, no one needed to make you want to ride a bike. No one had to convince you that it would serve you well later on in life, or that it would look good on your CV.

你从来没想过这些问题。你见过其他孩子骑自行车。你喜欢它，你也想这样做。

These questions never crossed your mind. You’d seen other kids riding bikes. You liked it and you wanted to do the same.

自 1687 年艾萨克·牛顿发表《自然哲学的数学原理》以来，人们就已知道了控制自行车运动的物理原理。在这篇开创性的论文中，他引入了万有引力和惯性原理。直到两个世纪后，自行车才被发明出来。如果你在牛顿生日时送他一辆自行车，他肯定不会骑。很有可能，他会觉得这个想法很愚蠢、很危险。他甚至可能证明在自行车上保持平衡在生理上是不可能的。但如果你向他展示如何做到这一点，他肯定会很感兴趣。

The principles of physics that govern the movement of bicycles have been known since 1687 and Isaac Newton’s publication of Philosophiae Naturalis Principia Mathematica, the groundbreaking treatise in which he introduced both universal gravitation and the inertia principle. The bicycle wouldn’t be invented until two centuries later. If you’d given one to Newton for his birthday, he would have refused to ride it. Most likely, he would have found the whole idea dumb and dangerous. He might even have gone so far as proving that it’s physically impossible to keep your balance on a bicycle. But if you had shown him how, he would have been intrigued.

数学欲望的秘诀

A Recipe for Mathematical Desire

如果我能展示出来，我脑子里进行的数学运算将会引起很多人的兴趣。但这么说毫无意义，因为我没有任何方法可以展示它。

If I could show it, the math that goes on in my head would intrigue a lot of people. But it serves no purpose to say so, since I don’t have any means of showing it.

当我试图让人们对数学产生兴趣时，我会采用不同的方法。我不会谈论我个人感兴趣的内容，而是选择最容易理解的科目，并专注于重现正确的情感历程。毕竟，这才是让我最初产生兴趣的原因：一种情感触发因素，一种非常幼稚的触发因素，就像有人“挑战”我，而我不想承认自己害怕一样。

When I try to get people interested in math, I approach it differently. Rather than talking about what interests me personally, I choose the most accessible subjects and focus on re-creating the right emotional journey. After all, this is what got me interested in the first place: an emotional trigger, a very childish one, as when someone “dared” me and I didn’t want to admit that I was scared.

我清楚地记得第一次看到人们风筝冲浪时的感受。我看到的景象似乎根本不可能实现。但与此同时——好吧，这显然是可能的。我看了很长时间。

I remember clearly how I felt the first time that I saw people kitesurfing. What I was seeing didn’t seem at all possible. But at the same time—well, it was apparently possible. I ended up watching it for a long time.

当我开始学习数学时，类似的事情也引起了我的兴趣。数学似乎太难、太抽象、太难以理解。数学似乎不是人类能够做到的。但与此同时，数学显然是可能的。

Something similar intrigued me when I got started in math. It seemed too hard, too abstract, too incomprehensible. It didn’t seem that doing math was humanly possible. And at the same time, it was apparently possible.

数学的难度，最初的震惊，只是情感旅程的第一部分。第二部分是深刻理解后产生的难以置信的惊奇感，一旦你发现它不仅是可能的，甚至很容易。它从一开始就很容易，只是你看不到它。

The difficulty of math, the initial shock, is only the first part of the emotional journey. The second part is the incredible feeling of wonder that arises from deep understanding, once you discover that not only is it possible, it is even easy. It was easy from the beginning, except that you couldn’t see it.

敬畏之后是魔法：这是激发数学欲望的强大秘诀，也是对课程中经过净化的数学的极大补充。数学不适合胆小的人。当我们隐藏它的可怕之处时，我们就会让它变得不那么令人向往。如果没有敬畏，就不会有魔法。

Awe and then magic: this is a potent recipe for mathematical desire and a great complement to the sanitized math from the curriculum. Math isn’t for the faint of heart. When we hide how scary it can be, we make it less desirable. If it wasn’t for the awe, there would be no magic.

测量无穷大

Measuring Infinity

普及数学的一个好课题是让你体验敬畏，然后体验神奇，而不会被数学所困扰专门的技术和语言。在这方面，格奥尔格·康托尔（1845-1918）的发现是完美的。

A good subject for popularizing math is one that allows you to experience awe and then magic without getting burdened down with specialized techniques and language. In that respect, the discoveries of Georg Cantor (1845–1918) are perfect.

无限的概念可以追溯到时间的黎明，几千年来，它一直象征着不可想象的事情。你可以长篇大论地谈论无限，但必须保持庄重的镇定和夸夸其谈的语气。这是一种冷静地不说话却听起来很深刻的方式。你不能随意而精确地谈论无限，就像你谈论数字 5 或与圆相交于两点的直线一样。

The concept of infinity goes back to the dawn of time and, for millennia, it had symbolized the unthinkable. You could talk at length about infinity, but only with a dignified composure and grandiloquent tone. It was a cool way of saying nothing while sounding profound. You were not allowed to talk about infinity in a casual and precise manner, in the same way you would talk about the number 5 or a straight line that intersects a circle at two points.

随意而精确地谈论无限就像登上月球：这是人类永远无法实现的完美典范。直到康托尔发现他可以做到这一点。更令人难以置信的是，在如此惊人的发现一个多世纪后，大多数人甚至都没有听说过它。

Being casual and precise when talking about infinity was like going to the Moon: it was the perfect example of something humankind would never achieve. Up until the day that Cantor found out he could do it. Even more incredible is that more than a century after such a spectacular discovery, most people haven’t even heard about it.

当我遇到一个不知道无穷大有许多种大小的人时，这几乎和遇到一个不知道你可以数到 5 以上的人是一样的。这给了我一个分享新闻的机会。

When I meet someone who doesn’t know there are many sizes of infinity, it’s almost the same as meeting someone who doesn’t know you can count beyond 5. It gives me a chance to share the news.

拿一个无限延伸的网格来说。我只会画出一小部分，但你会明白我在说什么：一张四面都无限延伸的网格纸。

Take a grid that stretches to infinity. I’ll only draw a bit of it, but you’ll see what I’m talking about: a piece of grid paper that goes to infinity on all sides.

在这个无限的网格中，有无限多个白色的方框。人们普遍理解这句话，觉得它简单而具体。

In this infinite grid, there are an infinity of white boxes. People generally understand this statement, which they find simple and concrete.

在一条无限长的直线上，也有无限多的点。这一点大家也都很清楚。大家也可以想象出这幅图。

On an infinite straight line, there are also an infinite number of points. That’s also generally clear to everyone. Everybody can picture this drawing as well.

网格中的方框数量是否比线上的点多？ 方框数量相同吗？ 线上的点数量是否比网格中的方框多？

Are there more boxes in the grid than points on the line? Is it the same number in each? Are there more points on the line than boxes in the grid?

我听到有人嘲笑这些问题。他们确信谈论无限是神秘主义者和神学家的专利。他们典型的反应是：“这个问题毫无意义”和“无限并不存在。”

I’ve heard people sneer at these questions. They were convinced that speaking about infinity was something reserved for mystics and theologians. Their typical reactions: “The question doesn’t make any sense” and “Infinity doesn’t exist.”

你不能同时拥有这两种情况：如果无穷大不存在，那么直线就不存在，或者它们只有有限数量的点。数学抽象的存在既不多也不少，与我们操纵的其他抽象一样。红色真的存在吗？电子真的存在吗？正义和自由真的存在吗？在第 18 章中，我们将讨论由大象这样务实而具体的概念所引起的难以克服的实际困难：从某种意义上说，大象并不存在。但这绝对不会阻止我们谈论大象，也不会阻止我们提出有关大象的精确问题并给出这些问题的答案。

You can’t have it both ways: if infinity doesn’t exist, then straight lines don’t exist, or they only have a finite number of points. Mathematical abstractions exist neither more nor less than other abstractions we manipulate. Does the color red really exist? Do electrons really exist? Do justice and freedom really exist? In chapter 18 we’ll talk about the insurmountable practical difficulties raised by a concept as pragmatic and concrete as that of an elephant: in a sense, elephants don’t really exist. That absolutely doesn’t stop us from talking about elephants, or from asking precise questions about them and providing answers to these questions.

康托意识到集合语言可以让你对有关无限性的问题给出精确的答案。

Cantor realized that the language of sets allows you to give precise answers to questions about infinity.

集合的概念非常古老。它自古以来就被非正式地使用，没有人反对它或花时间更仔细地研究它。你可以说“我家房子的集合”街道”或“你面前的一组苹果”或“一组整数”，每个人都明白它的意思。这个词是日常用语的一部分，并不被视为数学概念。

The concept of sets is very old. It had been used since ancient times, informally, without anyone objecting to it or taking the time to look at it more closely. You could speak of “the set of houses on my street” or “the set of apples in front of you” or “the set of whole numbers” and everyone understood what it meant. The word was part of everyday speech and wasn’t seen as a mathematical concept.

基于对集合构成的直观认识，康托尔创建了一个简单而富有表现力的数学词汇。他的定义并不比我们第 8 章中的触觉理论更复杂。借助这个词汇，我们可以非常精确地理解上面提出的问题，并给出一个既令人惊讶又清晰的答案：

Building on the intuitive idea of what constitutes a set, Cantor created a simple and expressive mathematical vocabulary. His definitions are no more complicated than our theory of touch in chapter 8. Thanks to this vocabulary, it’s possible to give a very precise meaning to the questions we asked above and provide a response as surprising as it is clear:

定理：直线上的点数比网格中的框数多。

Theorem: There are more points in the straight line than there are boxes in the grid.

最令人惊讶的是，从最初的定义到定理的证明，整个过程可以在一小时内向一个好奇的小学生解释清楚。换句话说，一个被认为不仅无法解决而且实际上不可想象的问题的解决方案就在我们眼前。它自古以来就存在，距离我们不到一小时的路程。

Most strikingly, the whole thing, from the initial definitions to the theorem’s proof, can be explained in under an hour to a curious primary school student. In other words, the solution of a problem that was deemed not just unsolvable but actually unthinkable was right before our eyes. It had been there since the beginning of time, less than an hour away from us.

我不是在开玩笑：我确实曾经在喝咖啡的时候向邀请我共进午餐的朋友的孩子们解释过这一点，他们确实很感兴趣。

I’m not joking: I’ve really explained it, over coffee, to the children of friends who’d invited me to lunch, and it really interested them.

以下是证明的粗略概述，简短而不完整。网格中无限数量的盒子被认为是可数的：你可以用整数给所有盒子编号（例如，从随机盒子开始，将其编号为 1，然后将周围的盒子编号为 2 到 9，依此类推，通过连续嵌套的方块编号直到无穷大）。康托发现，相反，直线是不可数的：其点的无限性如此之大，以至于你无法用整数对它们全部进行编号。为了证明这一点，他使用了一种今天被称为康托对角线论证的方法。

Here’s a rough outline of the proof, in a short and incomplete version. The infinity of boxes in the grid is said to be countable: you can number all the boxes with whole numbers (for example, by starting with a random box and numbering it as 1, then numbering the surrounding boxes as 2 through 9, and so on, numbering by successive nested squares up to infinity). Cantor discovered that, to the contrary, the straight line is uncountable: the infinity of its points is so large that you can’t number them all using whole numbers. To prove that he used a process that today is called Cantor’s diagonal argument.

而不是试图以书面形式向你解释细节（你你可以在维基百科上找到它们），我宁愿让你找一个可以亲自向你解释的人。正如我们在第 6 章中看到的，直接交流比阅读效率高得多。按照第 13 章中的建议，强迫自己问出所有想到的愚蠢问题：我保证你会有一些。

Rather than trying to explain the details to you in writing (you can find them on Wikipedia), I’d rather let you find someone who can explain it to you in person. As we’ve seen in chapter 6, direct communication is spectacularly more efficient than reading. Follow the advice in chapter 13 and force yourself to ask all the stupid questions that come to mind: I guarantee that you’ll have some.

康托尔本人也对他的发现感到震惊。他认为这是上帝直接赐予他的。关于他最意想不到的结果之一（直线上的点数与平面上的点数一样多），他在给一位朋友的信中承认：“我看到了，但我不相信！”

Cantor himself was shocked by what he had discovered. He thought that it had been directly sent to him from God. As regards one of his most unexpected results (there are as many points in a line as in a plane), he admitted in a letter to one of his friends, “I see it, but I don’t believe it!”

康托的成果如此新颖和具有开创性，以至于他不得不面对同时代人的怀疑。一位颇具影响力的数学家称他为“科学骗子”、“叛徒”、“腐蚀青年”。当他向一家领先的科学杂志提交一篇文章时，编辑恳求他撤回，因为他说这篇文章“早了大约一百年”。

Cantor’s results were so new and groundbreaking that he had to face the incredulity of his contemporaries. An influential mathematician labeled him a “scientific charlatan,” a “renegade,” a “corruptor of youth.” When he submitted one of his articles to a leading scientific journal, the editor implored him to withdraw it because, he said, it came “about one hundred years too soon.”

康托在生命的尽头，因饱受争议而陷入深深的沮丧。他在疗养院度过了人生的最后几年，并在贫困中死去。

At the end of his life, undermined by all the controversies, Cantor sank into a deep depression. He spent his final years in sanatoriums and died in poverty.

他的思想最终占了上风。自 20 世纪初以来，集合的概念已成为数学的核心。对于我们这一代人来说，想象没有集合的数学就像想象没有电的生活一样。

His ideas finally prevailed. Since the early twentieth century, the concept of sets has become central in mathematics. For someone of my generation, trying to imagine doing math without sets is like trying to imagine life without electricity.

抓住结

Grasping at Knots

当我想要解释什么是数学证明、它有什么用途以及你可以通过思想的力量构建的特殊确定性时，我喜欢使用取自结理论的例子。

When I want to explain what a mathematical proof is, what purpose it serves, and the special flavor of certitudes that you can construct through the power of thought, I like to use examples taken from knot theory.

在数学中，结是一种连接绳子两端的方式。例如，你可以将绳子的两端绑在一起，形成所谓的三叶结。

In mathematics, a knot is a manner of joining the two ends of a string. For example, you can tie two ends of a string together in what is called a trefoil knot.

绳子被假定为有弹性且牢不可破：只要你不解开它，你可以随意摆弄它而不会改变结。例如，你可以从上面打成三叶结的绳子开始，然后摆弄它得到这个：

The string is presumed to be elastic and unbreakable: as long as you don’t untie it, you can play with it as much as you like without changing the knot. For example, you can start with a string tied into a trefoil knot as above, and play with it to get this:

它仍然是三叶结，只是画法不同。如果你没有立即看到如何从第一幅图过渡到第二幅图，如果这让你有点头疼，请不要担心。这很正常。如果它让事情变得更容易，你可以用一根真正的绳子试试。

It’s still a trefoil knot, just drawn differently. If you don’t immediately see how you get from the first drawing to the second, if that gives you a bit of a headache, do not worry. It’s normal. If it makes things easier, you can try it with an actual piece of string.

将绳子绑在一起的最简单方法是打一个平结，或者解开一个结，如下所示：

The simplest way to tie a string together is in a trivial knot, or an unknot, as follows:

从某种意义上说，解开的结是结的“零”；它是一种实际上没有打结的结。你可以用不同的方式画它，例如：

In a way, the unknot is the “zero” of knots; it is a knot that isn’t really knotted. You can draw it differently, for example:

从视觉上看，绳子并没有真正打结，它仍然是解开的结。但是，还有其他方法可以画出一个简单的结，乍一看根本看不出你正在处理的是一个解开的结。例如，你可以像这样画一个：

It’s clear enough visually that the string isn’t really knotted, and that it’s still the unknot. But there are other ways of drawing a trivial knot where it’s not at all evident at first glance that you’re dealing with an unknot. You can draw one, for example, like this:

你能在不借助绳子或纸笔的情况下，在脑海中理清这幅画吗？尽管我受过处理这类东西的扎实训练，但我还是花了一段时间才弄清楚如何做到这一点。如果你能在几分钟内做到这一点，而无需任何特殊训练，那就太棒了！一旦你第一次就搞定了，下次做起来就会容易得多。

Can you untangle this drawing in your mind, without the aid of a string or pen and paper? Despite my solid training in manipulating these kinds of things, it took me a while to figure out how to do it. If you can do it in a few minutes, without any special training, bravo! Once you get it the first time, it gets much easier to do it again.

说实话，这个例子已经接近我想象能力的极限。而当我在脑中解开一幅更复杂的解结图时，这些极限就被大大超越了，就像这样：

To be honest, this example approaches the limits of my capacity for visualization. And these limits are greatly surpassed when it comes to mentally untangling a much more complicated drawing of an unknot, like this:

我不知道是否真的有人能在精神上解开这种东西，在视觉上“明显”地发现它只是解开的结。这个想法让我害怕，甚至想想它都会让我头疼。

I don’t know whether there are people really able to mentally untangle this kind of thing, to find it visually “obvious” that it’s just the unknot. The idea terrifies me and even thinking about it gives me a headache.

正是因为很难看出两幅不同的画代表同一个结，所以结理论是一个有趣的课题。

It’s precisely because it’s hard to see that two different drawings represent the same knot that knot theory is an interesting subject.

一旦你意识到有无数种方法可以画出相同的结，无论方法复杂与否，你也会意识到，没有什么可以保证两个画法不同的结实际上是不同的。例如，一个最初的合理问题是：三叶结真的与不结不同吗？

Once you realize that there are an infinite number of ways, more or less complicated, to draw the same knot, you also realize that nothing guarantees that two knots drawn differently are in fact different. An initial legitimate question is, for example: Is the trefoil knot really different from the unknot?

换句话说：你能否拿一根像三叶草一样打结的绳子，在不剪断绳子的情况下，将其扭转以解开结，然后将其放在桌子上形成一个圆圈？

In other words: can you take a string knotted like a trefoil and twist it to undo the knot, without cutting the string, and lay it on a table so that it forms a circle?

如果你尝试一下，你很快就会觉得这是不可能的。从实验上讲，你会说三叶草与非结节是不一样的。

If you try it, you’ll quickly get the impression that it’s not possible. Experimentally, you would say that a trefoil is not the same thing as an unknot.

我喜欢这个例子，因为它完美地说明了笛卡尔怀疑论以及印象和数学证明之间的根本区别。

I like this example because it’s a perfect illustration of Cartesian doubt and the radical difference between an impression and a mathematical proof.

值得尝试玩十分钟绳子，问自己以下问题：你对三叶结与普通结的区别的确定程度是多少——50%？80%？99%？99.99%？你能诚实地说没有怀疑的余地吗？

It’s worthwhile to really try playing with a string for ten minutes, asking yourself the following question: what’s your level of certainty that a trefoil knot really is different from a trivial knot—50 percent? 80 percent? 99 percent? 99.99 percent? Can you honestly say that there is no room for doubt?

我会更直白地问这个问题：你真的会拿你的生命来打赌吗？

I’ll ask the question more bluntly: would you, for real, bet your life on it?

你有什么保证说，不会有某种曲折的道路或不知从何而来的诡计，让你从一个三叶草结走到解开结呢？

What guarantee do you have that there’s not some winding road, some trick out of nowhere, that would allow you to go from a trefoil knot to the unknot?

这就像那些看似无法解决的难题。如果你有解决方案，你肯定知道有解决方案。但是如果你没有解决方案，你不知道是否没有解决方案，或者你只是还没有找到它。

It’s like with those puzzles that seem unsolvable. If you have the solution, you know for sure there’s a solution. But if you don’t have the solution, you don’t know whether there isn’t one, or that you just haven’t found it yet.

我们都认为三叶结与解开的结不同，但解开的结的复杂图画的存在告诉我们，我们不能总是依赖我们的第一印象。因为一根绳子看起来完全缠结了，并不意味着它真的缠结了。

We all have the impression that a trefoil knot is different from an unknot, but the existence of complicated drawings of an unknot shows us that we can’t always rely on our first impressions. Because a string looks like it’s completely tangled up doesn’t mean that it is really tangled up.

你可以想象一下，通过一系列非常复杂的操作，可以解开三叶结并回到原来的状态，而这些操作人类至今还不知道该如何去做。

You could imagine that it’s possible to untangle a trefoil knot and get back to an unknot by a series of maneuvers so complicated that no human has yet figured out how to do it.

乍一看，似乎不可能 100% 确定某件事。你必须考虑所有可能的情况的无限性扭动绳子的方法有很多。即使你花了十亿年的时间玩弄绳子，你也只能尝试有限数量的组合。

At first glance, it thus seems impossible to be 100 percent sure about something. You’d have to consider the infinity of all possible ways of twisting the string. Even if you spent a billion years playing with the string, you could try only a finite number of combinations.

数学推理的美妙之处在于能够操纵像结一样短暂易逝的物体，并对那些乍看起来不可能百分之百确定的问题给出百分之百确定的答案。

The beauty of mathematical reasoning is to be capable of manipulating objects as evanescent as knots and give answers of which you’re 100 percent sure to questions that at first seem impossible to answer with that degree of certainty.

我所说的“瞬息万变的事物”是指那些看起来无法通过语言严格操控的事物。打结的绳子不像一个整数。它不像你能用方程式表达的东西。它不是你觉得能够用语言表达的东西。

By “evanescent objects” I mean those objects that don’t appear to be made to be rigorously manipulated by language. A knotted string is not like a whole number. It doesn’t resemble something that you can tie up in equations. It’s not something you feel able to capture in words.

对于三叶结，绳子确实看起来像是打了结，如果不剪断绳子就无法解开。但你很难说出绳结到底在绳子上的哪个位置。它不在一个你可以用手指指着的特定位置。你可以感觉到绳结的存在，但你永远无法真正“抓住”它。

With a trefoil knot, it does seem like the string is knotted and that you can’t unknot it without cutting it. But you’d be hard pressed to say where exactly along the string the knot is situated. It’s not at a specific place you can put your finger on. You sense the presence of the knot, but you can never really “grasp” it.

当我还是学生的时候，我震惊地发现，用语言来把握结是可能的，而且可以百分之百地确定这一结果：

When I was a student, I was shocked to discover that it’s possible to grasp knots with language and to give a complete proof, 100 percent certain, of this result:

定理：三叶结与非结不同。

Theorem: A trefoil knot is distinct from an unknot.

该定理的证明概述在书末的“注释和进一步阅读”部分。

A proof of this theorem is outlined in the “Notes and Further Reading” section at the end of the book.

堆放橙子

Stacking Oranges

假装所有数学证明都可以向非专业人士简单解释，这是一个谎言。

It would be a lie to pretend that all mathematical proofs can be explained simply to nonspecialists.

最容易提出的问题有时是最难解决的。有很多问题很容易提出，但我们不知道如何回答。甚至有些问题很容易要求我们知道如何解决极其复杂的解决方案，而且似乎没有简单的答案。

The easiest problems to ask are sometimes the most difficult to solve. There are any number of problems that are easy to ask that we have no idea how to answer. There are even problems that are easy to ask that we know how to solve only with extremely complicated solutions, and for which there seems to be no easy answer.

开普勒猜想就是一个完美的例子。它为以下问题提供了初步答案：如何最好地堆放橘子？

Kepler’s conjecture is a perfect example. It provides a tentative answer to the following question: What is the best way to stack oranges?

伟大的天文学家和数学家约翰尼斯·开普勒（1571-1630）在 1611 年直觉地想到了解决方案，但未能证明其正确性。正如我们所见，数学家将您认为正确但无法提供严格证明的陈述称为猜想。

The great astronomer and mathematician Johannes Kepler (1571–1630) had an intuition for the solution in 1611 without, however, being able to prove that it was correct. As we’ve seen, a statement that you think is true but for which you can’t give a rigorous proof is what mathematicians call a conjecture.

在开普勒的时代，橘子是奢侈品。他用炮弹来描述这个问题。但显然这并没有改变答案。

In Kepler’s time, oranges were a luxury. He stated the problem in terms of cannonballs. But obviously that doesn’t change the answer.

更准确地说，问题是关于橙子，假设它们都是完美的球体，大小相同。如果你试图用这些橙子填满整个空间，你应该如何堆叠它们才能达到最大密度？

To be more precise, the question is about oranges presumed to be perfect spheres, all of the same size. If you tried to fill a space entirely with these oranges, how should you stack them to achieve maximum density?

使用相同的立方体，你可以轻松地填满整个空间而不留任何空隙，并获得 100% 的密度。但对于球体来说这是不可能的。

With identical cubes, you can easily fill up the entire space without leaving any gaps, and get a density of 100 percent. But that’s not possible with spheres.

在市场上，水果小贩通常将橙子堆成金字塔形。

At the market, fruit sellers usually stack oranges in a pyramid.

采用这种模式，密度约为 74.05%：空间中约 74.05% 为橙色，约 25.95% 为橙子之间失去了空间。开普勒猜想指出，这是最大密度，没有办法堆叠密度更大的橙子。

With this pattern, the density is about 74.05 percent: the space is filled about 74.05 percent with oranges, with about 25.95 percent of space lost between the oranges. Kepler’s conjecture states that this is the maximum density and that there’s no way of stacking the oranges that would be more dense.

直观来看，这似乎足够可信。但“足够可信”并不是数学家所说的证明。

Intuitively, that seems credible enough. But “credible enough” isn’t what mathematicians call a proof.

两个多世纪以来，没有人取得任何进展。高斯取得了第一个突破：1831 年，他证明，如果你要求橙子排列成规则、重复的图案，就像晶体中的原子一样，那么水果卖家堆放橙子的方式是最密集的。

For more than two centuries, no one made any progress. Gauss made the first breakthrough: in 1831 he proved that if you require that the oranges be arranged in a regular, repeating pattern, like atoms in a crystal, then the fruit seller’s way of stacking the oranges is the densest possible.

这是一个惊人的结果，但它并不能完全证明开普勒的猜想。从先验上讲，它并不排除可能存在一种非常奇特、毫无规律的堆叠方式，这种方式比有规律地堆叠十万亿个橙子的密度更大。

It’s a spectacular result but it doesn’t totally prove Kepler’s conjecture. A priori, it doesn’t exclude that there might exist a very bizarre way, with no regularity, of stacking a hundred thousand billion oranges that would be denser than stacking them regularly.

就我个人而言，我不知道该如何解决这个问题。这让我头晕目眩。

Personally, I have no idea how you might approach such a problem. It makes my mind spin.

高斯的突破之后，又过了150年，开普勒猜想才被彻底解决。第一个完整的解决方案是由1958年出生的美国数学家汤姆·黑尔斯于1998年给出的。

After Gauss’s breakthrough, it would be another 150 years before Kepler’s conjecture would be entirely solved. The first complete solution was given in 1998 by Tom Hales, an American mathematician born in 1958.

因此，这个猜想一直存在了 387 年。经过这么长时间的等待，习惯很难改变。有些人仍然称这个结果为“开普勒猜想”，而实际上应该说“黑尔斯定理”。

The conjecture thus remained open for 387 years. After such a long wait, habits are hard to break. Some people continue to say “Kepler’s conjecture” for the result itself, whereas one really should say “Hales’s theorem.”

我完全无法向一个小学生——或者任何其他人——解释 Hales 的证明，因为我从未仔细研究过它，而且我需要花费数年的时间才能理解它。

I’m completely unable to explain Hales’s proof to a primary school student—or anyone else, for that matter—given that I’ve never looked at it closely and it would take me years of work to try to understand it.

1998 年 9 月，汤姆·黑尔斯将其提交给数学年鉴，这是最负盛名的数学杂志之一。在被接受发表之前，科学文章通常会被送交一两位匿名审稿人以检查其有效性。黑尔斯的证明非常困难，数学年鉴不得不将其送交由十二位审稿人组成的委员会进行评估。

In September 1998 Tom Hales submitted it to Annals of Mathematics, one of the most prestigious math journals. Before being accepted for publication, a scientific article is normally sent to one or two anonymous referees to check on its validity. Hales’s proof was so difficult that Annals of Mathematics had to send it for evaluation to a committee of twelve referees.

甚至有必要组织国际会议，其唯一目的是试图理解该证明。四年后，评审委员会主席确认委员会成员“99% 确信”该证明的有效性：还不错，但离定理级别还差得很远。该论文最终于 2005 年 8 月被接受，此时距离提交已过去了近七年。

It was even necessary to organize international conferences whose sole aim was to try to understand the proof. After four years, the chair of the refereeing committee affirmed that the committee members were “99% certain” of the validity of the proof: not bad, but still far from theorem grade. The article was finally accepted in August 2005, nearly seven years after it had been submitted.

海尔斯证明的一个不同寻常之处是它部分基于计算机：虽然抽象的数学推理让海尔斯能够涵盖一般情况，但它忽略了有限数量的可能例外，这些例外必须单独研究。这些数以百万计的特殊配置只能通过蛮力来解决。正是这种深奥的数学和大量计算的结合使得证明如此难以评估。到目前为止，还没有人能够仅凭思维的力量来证明开普勒的猜想。

An unusual feature of Hales’s proof is that it is partly computer based: while an abstract mathematical reasoning allowed Hales to cover the general case, it left aside a finite number of possible exceptions that had to be studied separately. These millions of special configurations could be addressed only by brute force. It is this mixture of deep mathematics and massive computations that made the proof so difficult to assess. To this day no human has been able to prove Kepler’s conjecture through the power of thought alone.

开普勒的猜想与三维空间中的球体堆叠有关，但最佳球体堆叠问题可以在任何维度上提出。正如我们在第 9 章中提到的，对于任何整数n，都可以在n维空间中进行几何学研究。

Kepler’s conjecture bears on the piling of spheres in three dimensions, but the question of the optimal stacking of spheres could be posed in any dimension. As we mentioned in chapter 9, it is possible to do geometry in dimension n for any whole number n.

在二维空间中，这个问题很容易解决。二维空间中的球体是一个圆形。因此，这个问题变成了在桌子上排列硬币的最密集方式之一。最佳解决方案如图所示。

In two dimensions, the problem is easy enough to solve. A sphere in two dimensions is a circle. The problem thus becomes one of the densest way to arrange coins on a table. The optimal solution is shown in the figure.

事实上，这个结果比三维版本更容易证明。

This result is in fact much easier to prove than the version in three dimensions.

没有什么可以阻止你尝试超越三维。如果你从未研究过高维几何，那么承认有人敢于研究这类问题会让你感到有些害怕。

Nothing is stopping you from trying to look beyond three dimensions. If you’ve never done geometry in higher dimensions, it’s a bit intimidating to admit to yourself that there are people brave enough to take on these kinds of questions.

当解决三维问题需要近四百年的时间时，您可能会认为必须等待相当长一段时间才能看到更高维度的问题得到解决。

When it takes nearly four hundred years to solve a problem in three dimensions, you’d think you’d have to wait quite a while to see it solved for higher dimensions.

在四维空间中，我们仍然无法找到问题的答案。在五维、六维或七维空间中，我们同样无法找到答案。

The solution still isn’t known in four dimensions. Nor in five, six, or seven dimensions.

这就是为什么 1984 年出生的乌克兰数学家 Maryna Viazovska 的成果如此令人惊讶。2016 年，她开始使用新颖且非常优雅的技术解决八维问题。这已经是一个惊人的结果。三个月后，她与四位合作者一起，使用类似的方法解决了二十四维问题。由于这些突破，她于 2022 年获得了菲尔兹奖。

This is why the results of Maryna Viazovska, a Ukrainian mathematician born in 1984, created such surprise. In 2016, she began by solving the problem in eight dimensions using new and very elegant techniques. This already was a spectacular result. Three months later, along with four collaborators, she used a similar approach to solve the problem in twenty-four dimensions. For these breakthroughs, she received the Fields Medal in 2022.

这是目前唯一已知答案的三维以上维度。

These are for now the only dimensions above three for which the answer is known.

为什么我们可以在八维和二十四维中解决这个问题，但在四维或五维中却不能？原因在于，在八维和二十四维中，会发生一种不寻常的现象。这些维度中存在着特殊的数学对象，它们产生了令人难以置信的密集和谐的球体堆叠方式。在二十四维中，堆叠非常密集，以至于每个球体都与 196,560 个相邻球体接触。

Why can we solve the problem in eight and twenty-four dimensions, but not in four or five? The explanation is that in eight and twenty-four dimensions, an unusual phenomenon takes place. There are exceptional mathematical objects, specific to these dimensions, that give rise to incredibly dense and harmonious ways of stacking the spheres. In twenty-four dimensions, the stacking is so dense that each sphere is in contact with 196,560 neighboring spheres.

我以敬畏和魔力作为本章的开篇。敬畏这个词用来形容我对人类能够描述 24 维空间中堆叠球体的最佳方式这一消息的个人反应是再恰当不过了。我还没有亲身体验过理解证明的魔力，但我对别人能做到的喜悦感到很满足。

I began this chapter by speaking of awe and magic. To characterize my personal reaction to the news that it is humanly possible to describe the best way of stacking spheres in twenty-four dimensions, awe is an accurate word. I haven’t yet experienced the magic of understanding the proof myself, but I’m content enough with the joy that someone else did.

我想，对你来说，二十四维的概念本身就已经令人眼花缭乱了。数学的美妙之处在于，这种眩晕是可以克服的。

I imagine that for you the idea of twenty-four dimensions is already vertiginous in itself. The beauty of mathematics is that this vertigo can be overcome.

没有什么能阻止你理解二十四维的真正含义，也没有什么能阻止你理解在这些类型的空间中进行几何学的用处。所有这些：二十四维球体堆叠的几何学最显著的应用是 NASA 发送到太阳系外的旅行者 1 号和 2 号探测器的数据传输协议。高维几何学的基础知识在智力上是可以掌握的。你可以在几周内学会它们。最困难的是克服你的恐惧。

Nothing is stopping you from understanding what twenty-four dimensions really means, nor from understanding the usefulness of doing geometry in these types of spaces. There are practical uses for all this: the geometry of stacking spheres in twenty-four dimensions was most notably used in the data-transmission protocols for the Voyager 1 and 2 probes, sent by NASA outside of the solar system. The basics of geometry in higher dimensions is intellectually within your grasp. You can learn them in a few weeks. The most difficult thing is overcoming your fear.

显然，证明像汤姆·黑尔斯和玛丽娜·维亚佐夫斯卡那样的定理并不是每个人都能做到的。然而，体验一些敬畏之情和一些魔力却更容易实现，尽管最终付出的努力并不多。

Proving theorems like those of Tom Hales and Maryna Viazovska is obviously not within reach of everyone. However, getting to experience some of the awe and some of the magic is much more accessible, at the expense of an effort that in the end isn’t so much.

24 维橙子还可以用来说明另一个重要观点。直到最近，人们还经常听说女性在生理上无法进行几何学研究。她们被认为缺乏想象空间中物体的能力，这使得她们甚至无法阅读路线图。下次你听到这种话时，请随意提起 Maryna Viazovska。

Twenty-four-dimensional oranges can also be used to make another important point. Until recently, it was quite common to hear that women were biologically incapable of doing geometry. They were supposed to lack the ability to visualize objects in space, which made them incapable of even reading a road map. Next time you hear that kind of stuff, feel free to bring up Maryna Viazovska.

16

清晰度提高

16

Hyperlucidity

我最早的记忆是，我花了很长时间刻意想象，那是在我七岁的时候。一天晚上，我躺在床上，关掉灯，闭上眼睛，意识到只要稍加努力，我就能想象自己看到自己最喜欢的动画片。

My earliest memory of a deliberate and sustained effort to imagine something goes back to when I was seven. One night in bed, after having turned off the light and closed my eyes, I realized that with a bit of effort I could imagine seeing my favorite cartoon.

我没有和任何人谈论此事。

I didn’t speak about it to anyone.

我清楚地记得，我当时惊讶的心情就像昨天一样，我用当时的话来向自己描述这种现象：感觉就像我可以“在脑海中看电视”。

I remember like it was yesterday the amazement I felt and how I described the phenomenon to myself, in the words I used back then: it felt like I could “watch TV in my head.”

我能够想象出我从未见过的图像和场景。我甚至能够想象出新的情节。这给我留下了深刻的印象。我很喜欢，所以我当然会继续做下去。

I was able to visualize images and scenes I’d never seen. I was even able to imagine new episodes. It made a big impression on me. I loved it, so of course I continued doing it.

从那时起，清醒与睡眠之间的过渡以及醒来时的困倦感就在我的智力发展中扮演着重要角色。在我开始的每一个项目中，一旦它变得严肃起来，一旦我真正感兴趣，一旦我面临真正的挑战，它就会开始占据那个过渡空间。

The transition between waking and sleep, as well as sleepiness upon awakening, have since that time played a central role in my intellectual development. In every project I begin, once it gets serious, once I really get interested, once I’m confronted with a real challenge, it starts to occupy that liminal space.

记录我的梦境是我第一次真正尝试写作。大约 17 岁时，我开始对此感兴趣。起初，我发现直接写下我的梦境太难了，所以我试着一边大声念出梦境，一边记录下来。我的项目是收集梦境，就像你写日记或相册一样。

Transcribing my dreams was my first real attempt at writing. I was about seventeen when I started to get interested in this. At first, I found it too hard to directly write down my dreams, so I tried to record them while saying them aloud. My project was to collect them, like you would keep a journal or a photo album.

但发生了一些意想不到的事情，迫使我放弃了这个项目。夜复一夜，由于我试图记住并把它们付诸文字，我的梦想变得更加丰富和精确。

But something unexpected happened that forced me to give up the project. Night after night, as an effect of my trying to memorize them and put them into words, my dreams grew in richness and precision.

我的梦越来越好了。梦境太美好了，以至于我开始觉得烦人了。

I was dreaming better and better. I was dreaming so well that it started to become annoying.

起初，我只能记得一个梦的零碎片段。但两三周后，我每天都会讲述五六个不同的梦，每个梦都有完整的故事和足够多的细节，可以用文字或长时间的录音填满很多页。

At first, I could remember only small fragments, bits and pieces of a single dream. But after two or three weeks, I was recounting five or six different dreams every day, each with a complete story and enough details to fill up a lot of pages with writing or long minutes of recordings.

这变得令人难以承受。梦境的记忆占据了我太多的脑海和生活空间。我觉得这种内省最终会吞噬我。

It became overwhelming. The memories of my dreams were taking up too much space in my head and in my days. I felt like this exercise in introspection would end up consuming me.

然而，即使我停止记录，我仍继续记下它们。我并不是在寻找其中的意义。我只是想掌握把它们写下来的艺术。

Yet I continued to note them down, even after I’d stopped recording them. I wasn’t looking for meaning in them. I just wanted to master the art of writing them down.

对我来说，这就是写作的本质。从图像和感觉开始，寻找一种用文字表达它们的方法，使它们清晰而扎实。记录下当时的情况、所面临的风险、人和物体在空间中的位置、他们的动作和动作。尽可能简单而忠实地描述你所看到和感受到的。捕捉情绪、音乐、气味、质感。如果你能做到这一点，你就能做任何事情。

To me, this is the essence of writing. Starting from images and sensations and seeking a way to render them in words, to make them clear and solid. Transcribing the situations, what’s at stake, how people and objects are positioned in space, their actions and movements. Describing what you see and feel as simply and faithfully as possible. Capturing the moods, the music, the smells, the textures. If you can do that, you can do anything.

以我的经验来看，写下梦想是最接近数学写作的方式。

Writing down dreams, in my experience, is the closest you can get to mathematical writing.

我总是对那些说他们从不记得自己做过的梦的人感到震惊。有些人说他们从不做梦。这当然是不可能的。我们每晚都会做梦。

I’m always struck by the number of people who say they never remember their dreams. Some say they never dream. That’s of course impossible. We all dream, each night.

记住梦境并非与生俱来的能力。这是通过练习培养的能力。开始时有一些技巧，但要提高则需要一些技巧。你越是忠实地记录所见，你看到的就越多。

Remembering your dreams isn’t something you’re born with. It’s an ability you develop through practice. There are techniques to begin and techniques to get better. The more faithfully you learn to transcribe what you see, the more you see.

很长一段时间里，我都会在床头柜上放一本笔记本，中间放一支笔作为书签，用来记录我所有的梦境和夜里的所有想法。我甚至自学了如何在完全黑暗的环境下写作。

For a long time I kept a notebook on my bedside table, with a pen in the middle as a bookmark, to note down all my dreams and all the ideas that came to me during the night. I even taught myself how to write in complete darkness.

当我停止写下梦境时，我很快就失去了记忆的能力。当我强迫自己再次写下梦境时，即使只是一两个词，这种能力也会逐渐恢复。有时你必须坚持几个星期。最困难的部分是在很长一段时间无法做到之后试图重新捕捉梦境的第一部分。

When I stop writing down my dreams I quickly lose my ability to remember them. When I force myself to write them down again, even if it’s just a word or two, the ability gradually returns. Sometimes you have to keep it up for several weeks. The most difficult part is trying to recapture the first bit of a dream after a long period of not being able to do it.

成年后，我找到了自己的方式来利用入睡前的特殊精神状态。我学会了不再专注于让我心神不宁的事情，而是让自己沉浸在这些事情中。这种细微差别很微妙，但却至关重要。专注就是全神贯注地思考，寻找解决方案。它从来都不起作用，而且会让你无法入睡。沉浸在某件事中意味着以一种不带任何目的、不带任何兴趣的方式思考它。这几乎就像做梦一样。

As an adult, I’ve developed my own way of making use of the special state of mind just before falling asleep. Rather than focusing on subjects that preoccupy me, I’ve learned to simply let myself be filled with them. The nuance is subtle but fundamental. Focusing is thinking intensely, in search of solutions. It never works and it keeps you from sleeping. Being filled with something means contemplating it without a goal, in a decentered and disinterested manner. It’s almost like dreaming.

我可能错了，但在我看来，这种入睡技巧增加了我第二天早上醒来时产生有趣想法的机会。

I might be wrong, but it seems to me that this technique of falling asleep increases my chances of waking up the next morning with interesting ideas.

从另一个角度来看

From Another Point of View

在我的几何想象练习中，我最喜欢的练习是在入睡阶段进行的。

Among my exercises of geometric imagination, my favorite one is done during the phase of falling asleep.

我躺在床上，闭上眼睛，努力回忆我睡过的所有房间。我想象自己身处这些房间：床的大小和方向，墙壁和天花板的位置，门和窗户。我重现了过去躺在这些房间里的​​物理感觉。我试着用我的整个身心去体验它。我选择第一个想到的房间，然后是另一个，另一个，另一个，除了灵感之外没有特别的顺序。最有趣的是当我回忆起一个我早已忘记的房间时。

Lying in my bed, eyes closed, I try to remember all the rooms I’ve slept in. I imagine being in the room: the size and orientation of the bed, the location of the walls and ceiling, doors and windows. I re-create the physical sensation of lying down in one of these rooms from my past. I try to experience it with my entire being. I choose the first room that comes to mind, then another, and another, and another in no particular order other than that of inspiration. The most interesting thing is when I recall a room that I had long forgotten.

我喜欢这个练习，因为它简单又平和。对于初学者来说，这是一个很棒的练习。

I like this exercise because it’s easy and peaceful. It’s a great exercise for beginners.

正如我在第一章中所说，我很早就开始培养对空间和几何的直觉感知，但并没有发现它们与我在学校学到的数学有丝毫的关系。

As I said in chapter 1, I began early on to develop my intuitive perception of space and geometry, without seeing the least relationship to the math I was being taught at school.

时间已经过去很久了，我已经说不清它从什么时候开始的。那段时间，我学会闭着眼睛走路，记住墙壁和物体的位置，晚上躺在床上想象着动画片。

It’s so long ago that I can no longer say exactly when it started. The period when I was teaching myself to walk around with my eyes closed, memorizing the position of the walls and objects, was the same time I was imagining cartoons at night in bed.

我清楚地记得自己做这些事情时的心情。早在幼儿园的时候，我就因为近视而不得不戴眼镜。在小时候的想象中，我相信近视只是完全失明的第一步。我闭着眼睛四处走动，为再也看不见的那一天做准备。

I recall quite clearly my state of mind while I was doing those things. Early on, in kindergarten, I had to wear glasses because I was nearsighted. In my child’s imagination, I believed that being nearsighted was just the first step toward becoming completely blind. I walked around with my eyes closed to train myself for the day when I could no longer see.

我就是这样培养了自己的想象能力，也是这样开始学习几何的。后来，我想利用这种想象能力来训练自己能够随心所欲地从另一个角度看世界。

That’s how I developed my capacity for visualization, and that’s how I started learning geometry. Later on, I had the idea to use this capacity for visualization to train myself to be able, on command, to look at the world from another point of view.

我从字面意义上使用了“从另一个角度来看”这个短语。

I’m using the phrase “from another point of view” in a literal sense.

在我十二岁的时候，有一天，我们的美术老师要求我们画出铅笔盒的图案。我画的是铅笔盒内部的景象，里面的钢笔因为透视而变形了。没有人教过我如何利用透视来绘画，但我觉得这很自然：我只是在画我脑海中看到的东西。

I was twelve years old when one day our art teacher asked us to draw our pencil cases. I drew mine seen from inside, with gigantic pens deformed by the perspective. No one had taught me to draw using perspective, and yet it seemed natural to me: I was simply drawing what I saw in my head.

我为这幅画感到非常自豪，我记得它给我的同学留下了深刻的印象。

I was quite proud of this drawing and I remember it made a big impression on my classmates.

大约在我十五岁的时候，我们在课堂上学习了立体几何，也就是三维几何。直到那时我才意识到我的想象能力与我的朋友不同。这也许是我学生时代最超现实的记忆。

Around the time I was fifteen we’d studied solid geometry in class, that is, geometry in three dimensions. It was only then that I realized my capacity for visualization differed from that of my friends. It’s perhaps the most surreal memory of my school years.

我觉得课堂内容和练习完全没用，甚至毫无意义。我们好像又回到了幼儿园。如果老师只是举起一根手指问我们看到了多少根手指，我们只需回答“一根”，那也会给我留下完全一样的印象。

The class content and the exercises seemed to me so completely useless that they didn’t even make sense. It seemed like we were back in kindergarten. If the teacher had simply put one finger in the air and asked us how many fingers we saw, and we only had to answer, “One,” it would have made exactly the same impression on me.

我虽然不明白为什么我不明白，但我还是取得了最好的成绩。但我从朋友那里得知，立体几何对他们中的一些人来说是一门可怕的学科。这门课给他们带来了巨大的困难，以至于他们羞于谈论它。

I got the best grade without understanding how it was possible not to understand. But I knew from my friends that solid geometry was a frightening subject for some of them. It gave them such difficulties that they were embarrassed to talk about it.

我没有理由怀疑我转换视角的能力如此不寻常。我完全没有意识到我的几何过度训练程度。

I had no reason to suspect that my ability to switch viewpoints was so unusual. I was completely unaware of the extent of my geometric over-training.

如果你想练习切换观点，这里有一个很好的练习：

If you want to practice switching viewpoints, here is a good exercise:

1. 随机选择一个你周围的参考点，例如，房间里你对面的角落，或者你在街上行走时房子的窗户。

1. Choose a random reference point around you, for example, the corner opposite from you in a room, or the window of a house when you’re walking in the street.

2. 试想象一下，如果您从这个参考点朝您的方向看，您会看到什么。

2. Try to imagine what you’d see if you were looking in your direction from this reference point.

这不是二元对立的练习，有些人能做到，有些人做不到。这个练习对每个人来说都很难。一开始，你似乎永远看不到任何东西。但你一定会看到一些东西：一个模糊的形状、一个阴影或一团光斑，或者只是一个混乱而转瞬即逝的想法。你必须从那里开始。目标是提高图像的清晰度和分辨率，同时尽可能长时间地保持其存在。

It’s not a binary exercise in which there are those who can and those who can’t do it. The exercise is hard for everyone. At first, it seems like you’ll never be able to see anything. But you necessarily see something: a vague shape, a shadow or a blotch of light, or just a confused and fleeting idea. You have to start there. The goal is to increase the clarity and the resolution of the picture, while trying to maintain its presence for as long as possible.

我尝试过这项练习的多种变体。我甚至尝试过完全愚蠢的事情，完全无法完成的事情，只是为了好玩。

I’ve tried a number of variations of this exercise. I’ve even tried completely stupid things, things that were entirely undoable, just for fun.

有一次，我经常坐火车从巴黎去拉罗谢尔，我试图用视觉记住整个乡村，总是从火车的同一侧（西侧），然后在脑海中将其重新组合成一幅图像。最终的图像必须非常巨大：150 英尺高，300 英里长。这意味着要创造一幅宽度是高度 10,000 倍的图像，并填满整个画面。

At a time when I often took the train from Paris to La Rochelle, I tried to visually memorize all of the countryside, always from the same side of the train (the west side), and to reassemble it into a single image in my head. The resulting image would have had to be huge: 150 feet high and 300 miles long. It would have meant creating an image 10,000 times wider than it was tall and filling it entirely.

我从来没有成功过。

I never succeeded.

看见看不见的事物

Seeing the Unseen

在利用我的想象力从其他角度“看”世界之后，我养成了系统地尝试“看”所有应该在那里的事物的习惯，即使它们不可见。

After having used my imagination to “see” the world from other points of view, I got into the habit of systematically trying to “see” all the things that were supposed to be there, even though they weren’t visible.

让我用肥皂泡作为例子来解释这一点。

Let me explain this with an example, that of soap bubbles.

肥皂泡本质上和气球是一样的。肥皂水形成一层弹性膜，由内部气压膨胀。这就是为什么肥皂泡是球形的：为了容纳一定体积的空气，球形是表面积最小的形状。

A soap bubble is, in essence, the same thing as a balloon. The soapy water forms an elastic membrane, which is inflated by the internal air pressure. This is why the bubbles are spherical: to enclose a given volume of air, the sphere is the shape with minimal surface area.

当气泡形成时，你可以看到它在形成球形之前会波动几秒钟。在最初的波动过程中，很容易“感觉到”气泡表面是有弹性的。气泡波动缓慢，就像一个装满水的气球。我们对这种运动很熟悉，当它发生时，我们感觉我们可以“看到”气泡的物理特性。

When a bubble forms, you can see it undulate for a few seconds before it takes its spherical form. During this initial undulation, it’s quite easy to “sense” that the surface of the bubble is elastic. The bubble undulates slowly, like a balloon filled with water. We’re familiar with this movement, and as it takes place it feels that we can “see” the physical properties of the bubble.

我训练自己在气泡变成球形后继续“观察”气泡的弹性。我训练自己“观察”气泡内部的气压高于气泡外部的气压。

I’ve trained myself to continue to “see” the elasticity of the bubble once it has taken its spherical shape. I’ve trained myself to “see” that the air inside the bubble has a higher pressure than the air outside the bubble.

我在“看见”两边加了引号，因为我知道我的眼睛无法真正看见它。它是视觉的，但同时它又不是真正的视觉。我无法画出它。它只是一种位于我视野内的奇怪感觉，就像有什么东西被突出显示了一样。从某种意义上说，你可以说这是一种幻觉。但这是一种经过训练的幻觉，是经过构建和控制的幻觉。

I’m placing quotation marks around “see” because I know that I can’t really see it with my eyes. It’s visual, and yet at the same time it’s not really visual. I wouldn’t be able to draw it. It’s just a weird sensation that is located within my visual field, like if something had been highlighted. In a sense, you could say that it’s a hallucination. But it’s an educated hallucination, one that’s constructed and controlled.

当我观察一座桥时，我会看到应力线。我能看到桥的哪些部分受到压缩，哪些部分受到拉伸。

When I look at a bridge, I see stress lines. I see which parts of the bridge are subject to compression, and which ones are subject to tension.

只要集中注意力，我就能随时唤起这些感知。它们帮助我体验和理解世界。

I can call up these perceptions on command, with a bit of concentration. They help me experience and understand the world.

你也有类似的感觉。你可以“看到”绳子拉得太紧，即将断裂，或者气球充气过多，即将爆裂。你已经学会了“看到”物体的张力，就好像它是另一层增强现实，是嵌入你视野的补充信息，就像颜色一样，但位于另一层现实中。

You have similar sensations. You can “see” that a rope is stretched too tightly and is about to break, or that a balloon is overinflated and is about to burst. You’ve learned to “see” the tension of objects as if it was another layer of augmented reality, supplementary information embedded in your visual field, exactly like a color but located in another layer of reality.

这些例子可能看起来太普通了。让自己更多地了解这些类型的东西可能看起来幼稚而徒劳。然而，我觉得正是这种方法在我的一生中系统地应用，巩固了我的科学理解，并让我创作出原创的数学作品。

These examples might seem too ordinary. Educating yourself to see more and more of these types of things might seem childish and vain. Nevertheless, I have the feeling that it’s this approach, applied systematically throughout my life, that has solidified my scientific understanding and allowed me to produce original mathematical work.

那些我无法“看到”或“感觉到”的东西，即使我知道它应该是真实的，对我来说仍然具有特殊的地位。我不会忽略通过语言获得的外部信息，但我将其视为一种假设，并不完全相信它。

That which I’m unable to “see” or “feel,” even when I know it should be real, retains a special status for me. I don’t ignore the external information I get through language, but I treat it as a hypothesis, without entirely believing it.

如果我不明白为什么某件事应该是真的，我就会对其保持警惕。这种信息可能会在这种中间状态停留很长时间。可能是几个小时、几天、几周、几年，甚至几十年。

If I don’t see why something should be true, I’m wary of it. This kind of information can stay in this intermediate status for a long time. Maybe for a few hours, days, weeks, years, or even decades.

有时我会突然“理解”小时候学到的东西，而这些东西从那时起就一直处于这种中间状态。

Sometimes I suddenly “understand” things I was taught when I was a child, and that had since then remained in this intermediate status.

在我的高中地理课上，我们学到了森林砍伐会导致土壤侵蚀。这些信息是通过语言获得的，没有形象化，也没有真正理解，我完全不明白。它不引起我的兴趣，也没有说服我。

In my class of high school geography, we were taught that deforestation led to soil erosion. This information, which I received through language, without visualization and without really understanding it, went completely over my head. It didn’t interest me, and it didn’t convince me.

十年后，我终于明白了。我清楚地记得那一刻。当时我在一个数学会议上，在做演讲时，我感到很无聊，因为我什么都听不懂。我看着窗外的几棵树。我试着想象整个风景。我想象着这些树的整体，直到它们的根部。突然间，一切都说得通了。

A decade later, I finally understood. I remember the moment quite well. I was at a math conference, during a presentation where I was getting bored because I couldn’t understand anything. I looked out the window at some trees. I played at visualizing the scenery in its totality. I imagined the trees in their entirety, down to their roots. And all of a sudden, it all made sense.

从视觉上看，这很引人注目。树根网络形成了一种结构，可以保留土壤和岩石，就像钢筋混凝土中的钢筋一样，形成了一种强度惊人的复合结构。这解释了为什么土壤没有沿着山坡滑落。这也解释了我以前见过的另一个让我印象深刻的现象：移除树桩是多么的困难。

Visually, it was striking. The network of roots formed a structure that retained the soil and the rocks, like rebar in reinforced concrete, creating a composite structure with incredible strength. This explained why the soil wasn’t sliding down along the hillside. It also explained another phenomenon I’d seen before and that had struck me: of how unbelievably difficult it was to remove a tree stump.

另一个例子：我知道飞机能够飞，但我内心的一部分仍然拒绝相信，因为这看起来不正常，而且我无法直观地理解这是如何可能的。

Another example: I knew that planes were able to fly, but a part of me continued to refuse to believe it, because it didn’t seem normal, and I couldn’t intuitively understand how it was possible.

当我坐在飞机上，飞机沿着跑道加速准备起飞时，我内心的这一部分就出来了。我内心的一个小声音低声说：“真是个笑话，这东西飞不起来，太重了，我们永远也飞不起来。”

This part of me came out when I was seated in a plane accelerating down the runway for takeoff. A little voice inside me whispered something like, “It’s a joke, this thing can’t fly, it’s way too heavy, we’ll never take off.”

我想很多人都听过他们内心深处的这个声音至少，他们一生中会见过一次这样的人。但他们不敢承认，害怕自己看起来像个傻子。

I suspect that many people have heard this little voice inside their head, at least once in their life. But they don’t dare admit to it, afraid that they’ll look like fools.

我们社会上习惯于认为飞机起飞是正常的，并嘲笑有人会怀疑这一点。但我们中有多少人真正知道飞机是如何飞行的？

We’re socially conditioned to find it normal that airplanes can take off, and to laugh at the idea that someone could doubt it. But how many of us really know how planes manage to fly?

对飞机起飞能力产生怀疑并不愚蠢。这只是明智和独立思考的体现。

It’s not foolish to have doubts about the ability of planes to take off. It’s simply proof of good sense and independence of mind.

我的疑虑从未阻止我登上飞机。我并没有否认证据。我可以看到飞机能够飞行。这是我接受的，虽然有些不情愿，因为我从来没有得到过选择。我从实际的角度接受了它，但从感官的角度却不能。

My doubts never stopped me from boarding a plane. I didn’t go so far as to deny the evidence. I could see that planes were able to fly. It was something that I accepted, somewhat reluctantly, because I never had been offered a choice. I accepted it from a practical standpoint but not from a sensory standpoint.

当然，我必须以这种方式接受的事情比我真正能够理解的事情要多得多。

Of course, there are many more things I have to accept in this way than there are things that I’m really capable of understanding.

直到几年后，我才完全接受了飞机可以飞行的理念。我学会了如何用身体去感受它。为了做到这一点，我必须学会如何感受空气的密度和升力现象。我必须发现飞机比看上去要轻得多。我必须学会感受机翼、它们升起和弯曲的方式、它们的内部结构、它们如何连接到机身以及它们为什么不会断裂。

It’s only been a few years since I’ve completely accepted the idea that airplanes can fly. I learned how to feel it physically. In order to do so I had to learn how to feel the density of the air and the phenomenon of lift. I had to find out that airplanes are much lighter than they appear to be. I had to learn to feel the wings, the way that they lift up and bend, their internal structure, how they are attached to the fuselage and why they don’t break.

我的直觉最终让我发现这一切都很正常。飞机就像是我身体的延伸。

My intuition ended up finding all that normal. Planes became like an extension of my body.

在我的脊椎里

In My Spine

在克服了困难的数学科目并以此为职业之后，我对自己的几何直觉产生了很大的信心。

Having confronted difficult mathematical subjects and made a career of it, I developed a lot of confidence in my geometric intuition.

它尤其适用于大多数人不认为本质上是几何的事物。例如，当我得知发现弗洛勒斯人，然后是丹尼索瓦人，两种灭绝的物种时不久前，人类与我们共存的化石群让我感到很惊讶。这感觉很自然，原因在于我对所谓的“生命之树的几何形状”和“化石发现的几何形状”的直觉。

It comes into play notably about subjects most people don’t recognize as being geometric in nature. For example, when I learned of the discovery of Homo floresiensis, then Denisovans, two extinct species of humans that cohabitated with ours not so long ago, I wasn’t that surprised. It kind of felt natural, for reasons owing to my intuition of what I might call the “geometry of the tree of life” and “geometry of fossil discoveries.”

这并不意味着我就是魔术师。我的直觉是人之常情，会出错。我只是将我们每个人天生就具备的能力发挥到了极致。

This doesn’t make me a magician. My intuition is human and fallible. I’ve only pushed to its extremes an ability naturally present in all of us.

数学教会我，我们确实有可能在生活中取得进步和前进，而不必放弃童真的愿望，脚踏实地，只接受具体和明显的事物。

What mathematics taught me was that it was really possible to get ahead and move forward in life without giving up the childlike desire to remain down to earth, to accept only what is concrete and evident.

这真是出乎意料。我自己发明了这些视觉化和想象的练习，没想到它们会对我的生活产生如此大的影响。

That came as a surprise. I invented on my own these exercises of visualization and imagination, without expecting that they would have so much impact on my life.

在数学中，就像在许多其他领域一样，创造力只是理解的终极形式，它本身只是我们心理活动的自然产物。当我们强迫自己继续观察令我们畏惧的事物，直到它们最终变得熟悉和明显时，创造力就会出现。

In mathematics, as in many other fields, creativity is simply the ultimate form of understanding, which itself is but a natural product of our mental activity. It emerges when we force ourselves to continue looking at things that intimidate us until they finally become familiar and obvious.

掌握数学的方法有很多种。我们都是先从自己的长处和短处开始的。几何学从小就是我的强项之一，而且我有着丰富的视觉想象力。

There are many ways of grasping mathematics. We all start off with our strengths and weaknesses. Geometry has been one of my strong points since childhood and I have a long history of visual imagination.

我也有自己的弱点。例如，我从来没有真正对数字感兴趣，也从来没有花足够的时间去思考它们。话虽如此，通过重复和习惯，我确实设法对它们产生了一定程度的熟悉。我对它们的理解达到了一定的程度。但我从未觉得自己有能力在数论或算术方面发挥创造力。就像一个反手相对较弱、需要换手打正手的网球运动员一样，我已经陷入了养成尽可能避免使用数字的习惯，并寻找定量陈述的几何解释。

I also have my weaknesses. For example, I never really was interested in numbers, and never spent enough time thinking about them. That being said, through repetition and habit, I did manage to develop a degree of familiarity with them. I understand them up to a certain level. But I’ve never felt myself capable of being creative in number theory or in arithmetic. Like a tennis player whose backhand is relatively weak and who shifts to play his forehand, I’ve fallen into the habit of avoiding numbers whenever possible, and looking for geometric interpretations of quantitative statements.

如果我的主要兴趣是数字，我肯定会与数字建立更紧密的个人联系。这种直觉可能主要是视觉上的，也可能是完全不同的性质。最终这并不重要。所有数学家都以直觉的方式处理数学对象，但直觉有多种形式。

If my primary interest had been with numbers, I would certainly have developed a much stronger personal bond with them. This intuition might have been primarily visual, or of an entirely different nature. In the end it doesn’t matter that much. All mathematicians approach mathematical objects intuitively, but intuitions come in many shapes and forms.

我最大的弱点是无法理解复杂的符号，无法顺利理解包含大量符号和公式的推理。数学的整个领域都让我望而却步，尤其是分析。随着时间的推移，我失去了耐心，情况只会变得更糟。

My weakest point is my inability to find my way in complicated notations, to follow without stumbling reasonings that contain a lot of symbols and formulas. Entire areas of mathematics put me off because of this, especially analysis. It’s only gotten worse over time as I’ve lost patience with it.

某些科目我一开始并不擅长，但后来我设法进步了。为了理解 20 岁时给我带来很多困难的代数抽象结构，我发展了一种特殊的感觉直觉。

In certain subjects that at first I wasn’t very good at, I’ve managed to get better later on. To understand the abstract structures of algebra that caused me so many difficulties when I was twenty, I’ve developed a particular form of sensory intuition.

这些感觉非常强烈，但无法用语言表达。我通过自己身体内非视觉的运动感觉、张力和力场掌握了某些数学概念，仿佛我可以把自己传送到外部，从内部体验这些物体。

These are powerful sensations but impossible to put into words. I grasp certain mathematical concepts through nonvisual motor sensations, tensions and force fields within my own body, as if I could transport myself and experience these objects from the inside.

我感觉到这道数学题在我的脖子和脊椎里。

I feel this math in my neck and in my spine.

我知道这些词不太恰当，但我想不出更好的词了。这是一项想象力练习，在我努力提高这些科目时，它给了我很大的帮助。我看着一个物体，例如放在浴室里的一瓶洗发水，问自己以下问题：如果我的身体形状像瓶子，身体会有什么感觉？

I know these aren’t the right words, but I can’t think of better ones. Here is an exercise of the imagination that helped me a lot when I was trying to improve in these subject areas. I looked at an object, for example, a bottle of shampoo sitting in the bathroom, and asked myself the following question: if my body were shaped like the bottle, how would it feel physically?

通过对数学对象进行同样的练习，我能够理解它们。当我三十五岁的时候，这个技术nique 让我经历了职业生涯中最激烈的创作时期。

By practicing the same exercise with mathematical objects, I was able to understand them. When I was thirty-five years old, this technique led me to experience the most intense creative period of my career.

这一切都始于一次偶然的观察。有一天，我注意到，我问自己的一个关于 8 维空间中某些辫子的几何形状的问题可以很容易地翻译成范畴论的语言。两种截然不同的直觉之间意想不到的联系为我提供了一种新颖的方式来看待多年来一直困扰我的问题。

It all started with a casual observation. One day, I noticed that a question I was asking myself about the geometry of certain braids in dimension 8 could easily be translated into the language of category theory. The unexpected link between two very different intuitions offered a novel way of looking at problems I’d been struggling with for years.

我仿佛在大脑的两个区域之间架起了一座桥梁，而这两个区域此前从未相互沟通过。我突然顿悟了，然后又经历了一系列较小的余震。但这只是一个开始。我的数学想象力即将经历一次大规模的重构。

It was as if I’d built a bridge between two regions of my brain that up until then hadn’t communicated with one another. There was a big jolt of comprehension, then a series of smaller aftershocks. But it was just the beginning. My mathematical imagination was about to undergo a massive reconfiguration.

每天醒来我都会有新的想法。其中一些与我想要解决的问题有关（一个可以追溯到 20 世纪 70 年代的猜想），但其他一些则将我引向了完全不同的方向。我认为这些想法很棒，但我不得不放弃继续研究它们，因为不可能同时探索这么多线索。这实在是太多了。我试着做笔记，但我的理解进步速度比我写下来的速度要快。

Each day I woke up with new ideas. Some of them were tied to the problem I wanted to solve (a conjecture dating from the 1970s) but others took me in completely different directions. I thought they were beautiful, but I had to quit following them up because it was impossible to explore so many leads at the same time. It was just too much. I tried to take notes but my understanding progressed faster than my ability to write it down.

这种清醒状态持续了六周，此时我终于证明了这个猜想。

This state of hyperlucidity lasted six weeks, the time that I finished proving the conjecture.

我睡不着。我筋疲力尽。有一次，我早上 4 点醒来，突然想看看我十年前买的一本书（戴夫·本森的《表示与上同调》第二卷），当时我还没打开过。我在书架上找到这本书，拿起来，坐在地板上，一口气读了一百页，就像读一本漫画书一样。我以前从来没有读过这样一本数学书。如果我能读得这么快，那是因为我已经知道上面写了什么：就像我刚刚在梦里见过它一样。

I wasn’t able to sleep. I was exhausted. Once I woke up at 4 in the morning with the urge to look at a book I’d bought ten years earlier (the second volume of Representations and Cohomology by Dave Benson), which, at the time, I had barely opened. I found the book on a shelf, grabbed it, sat on the floor, and read a hundred pages in one go, as if it were a comic book. I’d never been able to read a math book like that before. If I was able to read it so fast, it was because I already knew what was written down: it was as if I had just seen it in my dream.

在这六周里，我感觉自己理解的新数学知识比我读博士以来十二年里理解的所有知识都要多。进展太快了，我都晕船了。身体上承受不住，我再也受不了了。我很痛苦。我希望这一切能停下来，这样我就能休息一下了。但它就是停不下来。就好像数学控制了我的大脑，违背我的意愿从我的脑袋里思考。

Throughout these six weeks, I had the feeling of understanding more new mathematics than everything I’d understood in the previous twelve years, since I had begun as a PhD student. It was going so fast that I was seasick. It was physically overwhelming and I could no longer cope. I was hurting. I wished that it would stop, so that I could get some rest. But it wouldn’t stop. It was like mathematics had taken control of my brain and was thinking from inside my head, against my will.

我生平第一次意识到极端数学是一项危险的运动。

For the first time in my life, I realized that extreme math was a dangerous sport.

17

掌控宇宙

17

Controlling the Universe

如果您要想象一个典型的年轻数学天才，我很清楚您会想到什么。

If you were to think of an archetypal young math prodigy, I have a pretty good idea of what would come to mind.

毫无疑问，他不会是那种整天和朋友聚会的班级小丑。他也不会是适应力强、善于交际、足智多谋、容易相处的人。他也不会是那种随和的人。

It undoubtedly won’t be the class clown who spends his time partying with friends. Nor a well-adjusted character, good at relationships, resourceful, and easy to live with. Nor the one who just takes it easy.

或许，你会想象像泰德·卡辛斯基这样的人。

Perhaps, instead, you’d imagine someone like Ted Kaczynski.

泰德·卡辛斯基 1942 年出生于芝加哥。他出色的数学能力很快被学校系统发现。在智商测试中得到 167 分后，他跳级了。几年后，他又跳级了。1958 年，16 岁的他被哈佛大学录取。1967 年，他获得数学博士学位，成为伯克利最年轻的助理教授。

Ted Kaczynski was born in Chicago in 1942. His exceptional math skills were soon spotted by the school system. After scoring 167 on an IQ test, he skipped a grade. A few years later he skipped another. He was accepted into Harvard in 1958, when he was sixteen years old. He completed his PhD in mathematics in 1967 and became the youngest assistant professor at Berkeley.

除了学业上的成功，泰德的生活很悲伤、孤独。那些在他年轻时认识他的人都说他情感上有问题，无法交流或建立真诚的关系。

Apart from his academic success, Ted led a sad and lonely life. Those who knew him in his youth described him as emotionally impaired, incapable of communicating or building genuine relationships.

在哈佛大学，与他同住宿舍的学生回忆起两个细节：他半夜吹长号的习惯，以及从门缝中飘出的腐烂食物的味道。

At Harvard, students he shared university housing with recalled two details: his habit of playing the trombone in the middle of the night, and the smell of rotting food that came from under his door.

泰德十五岁时，他很难与其他青少年交往，因此他更喜欢和八岁的弟弟大卫的朋友一起玩耍。

When Ted was fifteen, he had such difficulties socializing with the other teens that he preferred playing with the friends of his younger brother David, who was eight.

大卫·卡辛斯基回忆起他对情报部门的钦佩他很爱他的哥哥，但他也对他奇怪的行为感到惊讶。是什么让泰德交不到朋友呢？

David Kaczynski remembers his admiration for the intelligence of his older brother, whom he loved, but also his surprise at his strange behavior. What kept Ted from making any friends?

大卫八九岁的时候，有一天他问妈妈：“妈妈，泰迪怎么了？”

One day when David was eight or nine he asked his mother, “Mom, what’s wrong with Teddy?”

奇异现象的光谱

The Spectrum of Oddness

从本书一开始我就说过，我们需要摆脱对数学和数学家的刻板印象。我的想法没有改变。但这并不意味着这些刻板印象不存在，也不意味着它们毫无根据。

From the beginning of this book I’ve said that we need to get beyond the stereotypes of math and mathematicians. I haven’t changed my mind. But that doesn’t mean that these stereotypes don’t exist, or that they come out of nowhere.

这不是什么秘密。当你了解数学界时，你会立即注意到“奇怪”人物的数量。“奇怪”是一种很好的表达方式。奇怪程度是有的。有些数学家有点奇怪。有些数学家非常奇怪。还有一些数学家非常奇怪。在某些情况下，似乎超出了奇怪程度。只有一个词似乎合适：疯狂。

It’s no secret. When you get to know the mathematical community, one thing that immediately strikes you is the number of “odd” characters. “Odd” is a nice way of putting it. There are degrees of oddness. Some mathematicians are a bit odd. Others are quite frankly odd. Still others are spectacularly odd. In some cases, it seems to go beyond oddness. Only one word seems suitable: insanity.

事实上，这是数学家们经常谈论的话题。每个人都有几十个故事要讲。这些故事可能非常有趣，但它们太过夸张，令人难以相信。

It is in fact a recurring conversation topic among mathematicians. Everyone has dozens of stories to tell. These stories can be extremely funny, but they’re so caricatured that they’re hard to believe.

在所有我可以保证真实性的故事中，我只想分享一个：有一次我和一位数学家共进晚餐，他拿着一个装满他熟记的火车时刻表的垃圾袋四处闲逛。与他开始对话的最佳方式是询问周日下午从纽黑文到费城的交通方式。

Among all the stories whose authenticity I can vouch for, let me share just one: I once had dinner next to a mathematician who wandered about with a garbage bag filled with train schedules that he knew by heart. The best way to start a conversation with him was to ask the options for traveling from New Haven to Philadelphia on a Sunday afternoon.

打破刻板印象意味着要记住，大多数数学家都不是这样的。完全有可能在数学上达到最高水平，同时仍然保持“正常”。你可以在社交方面全面发展，热情开朗。你甚至可以成为一个魅力十足的领导者。幽默是数学家们比古怪更典型的特征。

Getting beyond the stereotypes means keeping in mind that the majority of mathematicians aren’t like that. It’s entirely possible to do mathematics at the highest levels and still be “normal.” You can be well rounded socially, warm, and open to others. You can even be a charismatic leader. Even more than oddness, humor is a fairly typical trait among mathematicians.

这并不意味着不会发生奇怪的事情，而且这种事情往往非常引人注目，尤其是在历史上一些最杰出的数学家中。在第 7 章中，我们谈到了亚历山大·格罗滕迪克的人生选择、他的极度孤独和禁欲主义。

That doesn’t mean that oddness doesn’t occur, often spectacularly, notably among some of the most prominent mathematicians in history. In chapter 7 we spoke about the life choices of Alexander Grothendieck, his extreme solitude and asceticism.

另一个引人注目的例子是格里沙·佩雷尔曼，我们在第 10 章中已经提到过他。他于 1966 年出生于列宁格勒，因在 2003 年证明了庞加莱在 1904 年提出的猜想而闻名。这是一项巨大的成就，其范围难以想象。

Another striking example is Grisha Perelman, whom we’ve already come across in chapter 10. Born in Leningrad in 1966, he is famous for having proven in 2003 the conjecture formulated by Poincaré in 1904. It was a massive achievement whose scope is hard to conceive.

佩雷尔曼不仅拒绝了菲尔兹奖，他还拒绝了克莱数学研究所于 2010 年颁发给他的百万美元奖金（庞加莱猜想位列该研究所的千禧年大奖难题之列，该难题被视为最难、对数学的未来最关键的七个难题之一）。

Perelman not only refused the Fields Medal, he also turned down the million-dollar prize awarded to him in 2010 by the Clay Mathematical Institute (the Poincaré conjecture figured on its list of the Millennium Prize Problems, the seven problems deemed to be the hardest and most pivotal for the future of mathematics).

2005 年，佩雷尔曼辞去了斯捷克洛夫研究所的职务。他不接受采访，很难知道他心里在想什么。他的个性如此引人注目，以至于网上流传着一些被盗的照片、虚假的采访和疯狂的谣言。

In 2005 Perelman resigned from his position at the Steklov Institute. He doesn’t give interviews and it’s difficult to know what is going on inside his head. His personality is so intriguing that there are stolen photos circulating on the internet, along with fake interviews and crazy rumors.

他很可能从未真正说过那句常被归于他的名言：“当我已经能够控制宇宙时，我要一百万美元会做什么？”

He most likely never really spoke the phrase often attributed to him: “What would I do with a million dollars when I can already control the universe?”

另一方面，另一句话来自一位权威人士：“金钱和名誉都不吸引我。我不想像动物园里的动物一样被人摆出来展览。我并不是数学英雄。我甚至没有那么聪明，这就是为什么我不想让每个人都看着我。”

Another statement, on the other hand, comes from a reputable source: “Money and fame don’t interest me. I don’t want to be put on display like an animal in the zoo. I’m not some mathematical hero. I’m not even as brilliant as all that, which is why I don’t want everyone looking at me.”

我们不得不钦佩佩雷尔曼的创造性智慧、他的智力、他坚定不移的决心以及他不可动摇的高尚心胸。

We can’t but admire Perelman’s creative intelligence, his mental prowess, his unfailing determination, and his incorruptible nobility of mind.

与此同时，还有一些令人深感不安的事情他的故事。听起来让人感到很不舒服。天才真的就应该是这样的吗？难道总是要以这种方式结束？真的不可能接触到其他人并找到与他们交流的方式吗？

And at the same time there’s something deeply troubling about his story. It comes across as a feeling of malaise. Is that really what genius is supposed to be? Does it always have to end this way? Is it really impossible to reach out to others and find a means of communicating with them?

我们忍不住觉得有些事情不对劲。我们认为，如果佩雷尔曼拒绝剪指甲，而且在年近六十的年纪还继续和母亲住在圣彼得堡的一间小公寓里，那一定是因为他身上出了问题。

We can’t help feeling that something isn’t right. We think that if Perelman refuses to cut his fingernails and, at nearly sixty years old, he continues to live with his mother in a small apartment in Saint Petersburg, it’s because something’s wrong with him.

这是有可能的。但是我们知道什么呢？佩雷尔曼是这个星球上最聪明的人之一。他从未伤害过任何人，他有权随心所欲地生活。我们没有资格评判他。

It’s possible. But what do we know? Perelman is one of the most brilliant minds on the planet. He’s never done anyone any harm and has the right to live as he pleases. We’re in no position to judge him.

是数学本身让人们变得“奇怪”吗？我不这么认为。我更愿意说，数学对那些已经有点“奇怪”的人来说是受欢迎的。

Is it mathematics itself that makes people “odd”? I don’t think so. I’d say rather that math is welcoming to people who are already a bit “odd.”

这是少数即使无法融入也能取得巨大成就的职业之一。对于那些已经有点“古怪”、“与众不同”、在社会上不自在的人来说，这可能是一条通往社会化和自​​我实现的道路。（我倾向于把自己归为这一类。）

It’s one of those rare careers where it’s possible to accomplish great things even if you can’t fit in. For people already a bit “odd,” “different,” not at ease in society, it can be a path toward socialization and fulfillment. (I tend to put myself in this category.)

对于大多数人来说，数学并不是那么危险。

To most people, math isn’t really that dangerous.

然而，数学有一个重大禁忌，即在特定情况下，数学可能会产生灾难性的副作用。如果根据历史上令人不安的频繁发生的例子来判断，数学似乎在某些情况下会滋生和放大一种特殊的心理病态：偏执狂。

There is, however, one major contraindication, a specific context where math can have catastrophic side effects. If you judge by the examples that occur throughout history with troubling frequency, mathematics seems, in certain cases, to feed and amplify a particular mental pathology: paranoia.

黑暗之心

Heart of Darkness

在怪异程度排行榜上，泰德·卡辛斯基名列前茅。

On the spectrum of oddness, Ted Kaczynski ranked near the top.

创造力与古怪并不成正比。早熟也不一定预示着辉煌的职业生涯。泰德·卡辛斯基从来就不是一位伟大的数学家。他微薄的科学成果并没有什么特别的价值。

Creativity doesn’t come in proportion to oddness. And precocity doesn’t always foreshadow a brilliant career. Ted Kaczynski was never a great mathematician. His meager scientific output doesn’t have any particular merit.

1969 年 6 月 30 日，27 岁的泰德·卡辛斯基突然辞去了伯克利大学的职务，没有给出任何解释。两年后，他搬到了蒙大拿州一个偏僻地区，在森林里自己建造的小屋里居住。

On June 30, 1969, at twenty-seven years old, Ted Kaczynski abruptly resigned from his position at Berkeley without giving any explanation. Two years later, he went to live in a cabin he built himself in the forest in an isolated part of Montana.

他选择独自生活，没有自来水，没有电。

He chose to live alone, without running water or electricity.

从 1971 年开始，他在日记中谈到了自己的犯罪意图：“我强调我的动机是个人复仇。我不会假装任何形式的哲学或道德辩护。”

Beginning in 1971, in his journal, he talks about his criminal intent: “I emphasize that my motivation is personal revenge. I don’t pretend any kind of philosophical or moralistic justification.”

随着时间的推移，他的言论发生了变化。后来，卡钦斯基声称他想发起一场革命。但他始终保持着一种似乎发自内心的感觉：对科学机构和官僚机构的仇恨，他认为这些侵犯了他的个人自由和一般自由：“我的野心是杀死一名科学家、大商人、政府官员或类似的人。我也想杀死一名共产党员。”

His discourse changed over time. Later on, Kaczynski would claim that he wanted to start a revolution. But throughout he maintained a feeling that seemed visceral for him: a hatred of the scientific establishment and bureaucracy, which he saw as attacking his personal freedom and freedom in general: “My ambition is to kill a scientist, big businessman, government official or the like. I would also like to kill a Communist.”

多年来，泰德·卡辛斯基在自己逐渐熟悉的森林中独自漫步，制定了自己的计划。他要报复工业社会给他带来的苦难，报复它为了修建新路而无情砍伐树木造成的破坏。

Over the years, during long solitary walks through the forest that he came to identify with, Ted Kaczynski developed his plan. He would get his revenge on industrial society for all that it had made him suffer, and for the violence it inflicted on the trees that it heartlessly cut down to build new roads.

凭借着方法和决心，他越来越陷入了令人难以置信的杀戮网。

With method and determination he fell further and further into an unbelievable murderous web.

直到 1996 年，在经过 FBI 历史上最长、最昂贵的调查后，他才被抓获。这次调查持续了 17 年多，动用了多达 150 名全职特工。

He wasn’t captured until 1996, after the longest and most costly investigation in the history of the FBI. It had lasted more than seventeen years and employed up to 150 full-time agents.

1998 年，泰德·卡辛斯基开始在科罗拉多州弗洛伦斯的超级监狱服刑，连续八个终身监禁，不得假释。该监狱是美国最安全的监狱。与他同狱的囚犯包括 9/11 恐怖分子扎卡里亚斯·穆萨维和墨西哥毒枭埃尔·查波等人。

In 1998, Ted Kaczynski began serving eight consecutive life terms without the possibility of parole at the super-maximum facility in Florence, Colorado, the most secure prison in the United States. His co-detainees included the likes of Zacarias Moussaoui, one of the 9/11 terrorists, and El Chapo, the Mexican drug baron.

他于 2023 年去世，疑似自杀。

He died in 2023 in an apparent suicide.

他的故事很悲观，但值得讲述。它可以教会我们关于理性的一些基本知识：理性的力量、局限性、危险，以及使用它的正确方法和错误方法。

His story is bleak but it needs to be told. It can teach us something fundamental about rationality: its power, limitations, dangers, and the good and bad ways to use it.

大学炸弹客

Unabomber

1979 年 11 月 15 日，美国航空 444 号航班从芝加哥飞往华盛顿特区。飞行途中，乘客听到一声闷响。机舱内充满刺鼻的烟雾，氧气面罩掉落。烟雾非常浓，甚至进入了面罩内。

On November 15, 1979, American Airlines flight 444 left Chicago for Washington, DC. Midway through the flight the passengers heard a muffled sound. The cabin filled with acrid smoke and the oxygen masks fell. The smoke was so dense that it got inside the masks.

飞行员成功紧急降落。12 人因吸入烟雾入院。地面上的初步调查结果毫无疑问：飞机上有一枚炸弹，如果炸弹正确运作，飞机将在空中被炸毁。

The pilot was able to make an emergency landing. Twelve people were hospitalized with smoke inhalation. On the ground, the initial findings left no doubt: there had been a bomb in the plane which, if it had worked correctly, would have obliterated it midair.

人们很快就与芝加哥附近的西北大学校园内遗留的两个装有炸弹的包裹建立了联系。

A link was quickly established with two booby-trapped packages left on the campus of Northwestern University, near Chicago.

这是一系列类似事件中的第一起。

This was the first in a very long series of similar incidents.

1980 年 6 月 10 日，美国联合航空公司总裁珀西·伍德 (Percy Wood) 因收到寄往芝加哥郊区森林湖 (Lake Forest) 家中的包裹炸弹而受重伤。1981 年，犹他大学校园内一枚炸弹被拆除。

On June 10, 1980, Percy Wood, president of United Airlines, was seriously injured by a package bomb sent to his home in Lake Forest, a suburb of Chicago. In 1981, a bomb was defused on the campus of the University of Utah.

袭击一直持续到 1995 年。16 枚炸弹导致 3 人死亡，23 人受伤。袭击目标是大学（伯克利的一栋建筑两次遭到袭击）和与航空、工业或技术相关的企业（其中包括波音办公室、电子产品商店和木材行业的说客）。

The attacks continued until 1995. Sixteen bombs resulted in three people dead and twenty-three wounded. The targets were universities (a building in Berkeley was targeted twice) and businesses having to do with aviation, industry, or technology (among them the Boeing offices, electronics stores, and a lobbyist for the timber industry).

在这起难以破解的连环恐怖案件中，侦查人员根据最细微的线索进行追踪。

In this indecipherable case of serial terrorism, the investigators followed up on the slightest of clues.

尽管有些炸弹上印有首字母缩写 FC，后来发现这是“自由俱乐部”的缩写，但联邦调查局仍然坚信这些袭击背后只有一个人。特工们对这名男子进行了心理分析，他非常憎恨大学和空中交通。他还痴迷于木材和森林。这种迷恋体现在目标的选择上（木材行业、珀西伍德、森林湖），也体现在制造炸弹所用的材料上。有些炸弹里有树皮碎片；有些则被伪装成原木。

Although some devices carried the initials FC, which was later revealed to stand for “Freedom Club,” the FBI remained convinced there was only a single person behind the attacks. Agents constructed a psychological profile of a man fed by a fierce hatred against universities and aerial transport. He was also obsessed with wood and forests. This fascination manifested itself in the choice of targets (the timber industry, Percy Wood, Lake Forest) as well as in the materials used to make the bombs. Some contained pieces of bark; others were camouflaged to resemble logs.

联邦调查局和媒体将他称为“大学炸弹客”，即“大学和航空公司炸弹客”的缩写。

The FBI and the media dubbed him the “Unabomber,” short for “University and Airline Bomber.”

至于炸弹本身，调查人员面临很多困难。通常情况下，可以让留在现场的每一块碎片“说话”：通过分析像钉子一样小的东西，他们也许能够确定制造商和可能出售的地方。

As for the bombs themselves, they posed a lot of difficulties for investigators. Ordinarily, it’s possible to make each fragment left at the scene “speak”: by analyzing something as small as a nail, they might be able to identify the manufacturer and places where it might have been sold.

问题是，大学炸弹客使用的钉子全都是手工制作的。没有任何线索可循，也没有指纹。炸弹的部件全都经过精心打磨。

The problem was that the nails used by the Unabomber were all made by hand. There were no clues to follow, no fingerprints. The components of the bombs had all been carefully sandpapered.

恐怖分子是一个耐心而又细心的人，他不怕从头开始构建一切。

The terrorist was someone patient and meticulous who was not afraid of building everything from scratch.

调查人员不得不等到 1995 年才开始调查此案。那一年，大学炸弹客向《纽约时报》、《华盛顿邮报》和《阁楼》杂志寄送了一份打字稿，并附上了一封信他说，如果这篇文章被发表，他将停止攻击。根据联邦调查局的建议，《纽约时报》和《华盛顿邮报》于 1995 年 9 月 19 日发表了这篇文章。

The investigators had to wait until 1995 for an opening in the case. That year the Unabomber sent a typed manuscript to the New York Times, the Washington Post, and Penthouse along with a letter in which he said he would give up his attacks if the text was published. Following the recommendation of the FBI, the New York Times and the Washington Post published it on September 19, 1995.

大学炸弹客的宣言《工业社会及其未来》结构严谨，论证严谨。宣言共包含 1 至 232 段，是对现代社会以及技术对我们生活的控制的激进批判。大学炸弹客认为，没有什么值得挽救的，所以整个系统都应该被摧毁。

The Unabomber manifesto, “Industrial Society and Its Future,” is meticulously constructed and argued. Constituting paragraphs numbered from 1 to 232, it’s a radical critique of modern society and the hold that technology has on our lives. According to the Unabomber, there’s nothing worth saving, so the entire system needs to be torn down.

这些言论有时很中肯，有时却很荒谬。有些段落透露出你不会想到典型恐怖分子会关注的重点：“确实，人们可以对科学知识的基础提出严肃的问题，以及如何定义客观现实的概念（如果有的话）。但很明显，现代左派哲学家不仅仅是头脑冷静的逻辑学家，系统地分析知识的基础。”

The statements are at times relevant and at other times absurd. Certain passages betray preoccupations that you wouldn’t expect from your typical terrorist: “It is true that one can ask serious questions about the foundations of scientific knowledge and about how, if at all, the concept of objective reality can be defined. But it is obvious that modern leftish philosophers are not simply cool-headed logicians systematically analyzing the foundations of knowledge.”

当大卫·卡辛斯基读到这段话时，他被“头脑冷静的逻辑学家”这句话震撼了。他想起了弟弟泰德在一封信中用一模一样的词语表达了同样的事情。

When David Kaczynski read this passage, he was struck by the phrase “cool-headed logicians.” He remembered a letter in which his brother Ted used exactly the same words to say exactly the same thing.

怀疑弟弟是美国头号通缉犯的戴维·卡辛斯基面临严重的道德困境。他是应该告发弟弟并冒着被判处死刑的风险，还是保持沉默并冒着成为可能杀害新受害者的帮凶的风险？

Suspecting that his brother was the most wanted criminal in the United States, David Kaczynski was confronted with a terrible moral dilemma. Should he turn in his brother and risk seeing him sentenced to death, or stay quiet and risk becoming an accomplice to the possible murder of new victims?

经过长时间的思考，他选择向联邦调查局求助。这导致泰德·卡辛斯基于 1996 年 4 月 3 日被捕。

After lengthy reflection, he chose to speak to the FBI. This led to the arrest of Ted Kaczynski on April 3, 1996.

“粗略估计真相”

“A crude approximation to the truth”

1996 年初，当调查人员面对这一意外线索时，他们试图评估其可信度。一些这件事让他们很困扰。泰德·卡辛斯基的生平与他们心中的“大学炸弹客”形象不符。有些元素是吻合的，尤其是生存主义生活方式，但他的学术成就和数学家背景却令人惊讶。

At the start of 1996, when the investigators were confronted with this unexpected lead, they sought to evaluate its credibility. Something troubled them. Ted Kaczynski’s biography didn’t correspond to the image they had of the Unabomber. Certain elements matched, notably the survivalist lifestyle, but his degree of scholarly achievement and his past as a mathematician were a surprise.

为了解决这种不确定性，联邦调查局决定秘密咨询当时担任伯克利数学科学研究所所长的比尔·瑟斯顿。

To address the uncertainty, the FBI decided to secretly consult Bill Thurston, who was at the time director of the Mathematical Science Research Institute at Berkeley.

读完这份宣言后，瑟斯顿毫不怀疑。他立刻明白了，这是一位数学家写的。

After he read the manifesto, Thurston had no doubts. It was immediately clear to him that it had been written by a mathematician.

我不知道瑟斯顿依靠哪些因素得出了这个结论。我自己重做了一遍（虽然一旦你知道故事的结局，就会容易得多），我最震惊的是最后几段。在用几十页的篇幅宣称确定性之后，卡辛斯基突然指出了他花了 25 年建造的这座疯狂大厦的脆弱性：

I don’t know which elements Thurston relied on to reach this conclusion. Redoing the exercise myself (although it’s much easier once you know the end of the story), I was above all struck by the last paragraphs. After having proclaimed certitudes for dozens of pages, Kaczynski suddenly points to the fragility of the delirious edifice he had spent twenty-five years constructing:

最后说明

FINAL NOTE

231. 在本文中，我们做出了不精确的陈述……我们的一些陈述可能完全是错误的……当然，在这种讨论中，人们必须严重依赖直觉判断，而直觉判断有时可能是错误的。因此，我们并不声称本文表达的只是对事实的粗略近似。

231. Throughout this article we’ve made imprecise statements . . . and some of our statements may be flatly false. . . . And of course in a discussion of this kind one must rely heavily on intuitive judgment, and that can sometimes be wrong. So we don’t claim that this article expresses more than a crude approximation to the truth.

真相的概念是卡钦斯基关注的核心。他指责现代哲学家领导了一场“对真相和现实的攻击”。

The concept of truth is at the center of Kaczynski’s preoccupations. He blames modern philosophers for having led an “attack against truth and reality.”

但真相是什么？卡钦斯基怎么能说他的宣言只是粗略的近似，却又认为以这个真相的名义杀人是合理的？他是不是想暗示他个人可以接触到真相，尽管他没有在宣言中完全写下来？

But what is truth? How could Kaczynski say that his manifesto was only a crude approximation, and yet find it reasonable to kill in the name of this truth? Did he want to suggest that he personally had access to the truth, even though he didn’t entirely set it down in his manifesto?

尽管卡钦斯基一直意识到自己的论点很薄弱，但他似乎坚信自己是对的，而其他人都是错的。他从来没能完全证明这一点，但这并不让他担心：对他来说，一切都很清楚。他编造了自己的真理和确定性，并将它们组织成一个连贯的体系，其他人和其他观点在其中没有立足之地。

Despite all the while recognizing the weakness of his argument, Kaczynski seemed convinced he was right and the rest of the world was wrong. He was never able to entirely prove it, but that didn’t worry him: for him, everything was clear. He had fabricated his own truths and certainties, and organized them into a coherent system in which other people and other opinions had no place.

当他从媒体上得知，他所投掷的炸弹第一次炸死了一个人，而且受害者被炸成了碎片时，他在日记中写下了庆祝的文字：“太棒了。用人道的方式杀死一个人。他可能根本感觉不到任何感觉。”

When he learned in the press that, for the first time, one of his bombs had killed someone and that the victim had been torn to pieces, he wrote a celebratory entry in his journal: “Excellent. Humane way to eliminate somebody. He probably never felt a thing.”

这一切让人感到不安。如果卡钦斯基利用数学家的技巧达到这种令人恐惧的自我激进程度，情况会怎样？

There’s a troubling impression that arises from all this. What if Kaczynski had used techniques of mathematicians to arrive at this frightening level of self-radicalization?

试图重塑直觉的技术可能会带来危险，这并不奇怪。毕竟，误用菜刀甚至土豆削皮刀都会让你被送进急诊室。

It shouldn’t come as a surprise that techniques that seek to reprogram your intuition could turn out to be dangerous. After all, misusing a kitchen knife or even a potato peeler can already send you to the emergency room.

据审判前对他进行检查的精神病医生称，泰德·卡辛斯基患有偏执型精神分裂症。卡辛斯基认为，这一诊断是政治迫害。他认为自己心智健全。他的律师想为他辩护，但卡辛斯基拒绝了，而是选择认罪。

According to the psychiatrist who examined him before his trial, Ted Kaczynski suffered from paranoid schizophrenia. According to Kaczynski, this diagnosis was political persecution. He believed himself to be of sound mind. His lawyers wanted to enter an insanity defense but Kaczynski refused, instead choosing to plead guilty.

偏执狂与数学推理关系密切。从某种意义上说，它们是邪恶的孪生兄弟。有些人甚至很难区分它们。不过，有一种简单的方法可以区分它们，我们稍后会谈到这一点。

Paranoid delirium is a close relation to mathematical reasoning. In a way it’s the evil twin. Some people even have difficulty telling them apart. There is, however, an easy way to distinguish them, which we’ll get back to.

“数学家该做什么？”

“What’s a mathematician to do?”

2010 年，即去世前两年，比尔·瑟斯顿似乎仍然专注于泰德·卡辛斯基的悲惨命运。

In 2010, two years before his death, Bill Thurston still seemed preoccupied with the tragic fate of Ted Kaczynski.

他花时间写了一篇长文，回答了数学社区协作网站 MathOverflow 上提出的一个问题。这个问题来自一个名叫 Muad 的用户，他显然是一个缺乏自信的年轻学生。文章的标题是“数学家该做什么？”

He took the time to write a long response to a question asked on MathOverflow, a collaborative site for the math community. The question came from a user named Muad, apparently a young student lacking in self-assurance. It’s entitled “What’s a mathematician to do?”

穆阿德问自己能为数学做出什么贡献。他觉得“数学是由高斯或欧拉这样的人创造的”，你可以尝试理解他们的工作，但这种理解不会带来任何新的东西。他担心的是，像他这样的人，普通人，那些没有任何“特殊天赋”的人，没有什么新东西可发现。

Muad asks how he can contribute to mathematics. He has the feeling that “mathematics is made by people like Gauss or Euler,” whose work you can try to understand without this understanding leading to anything new. His fear is that there’s nothing new to discover for people like him, normal people, those who don’t have any “special talent.”

大多数数学系的学生都会在某个时候经历过类似的感受。瑟斯顿的回答提供了一个彻底的视角转变：

Most math students experience similar feelings at one point or another. Thurston’s response offers a radical change of perspective:

数学的产物是清晰和理解，而不是定理本身。

The product of mathematics is clarity and understanding. Not theorems, by themselves.

世界并不会因为清晰度和理解力过剩而遭受损失（说得委婉些）。

The world does not suffer from an oversupply of clarity and understanding (to put it mildly).

数学的真正满足感在于向他人学习并与他人分享。我们所有人都对一些事情有清晰的理解，但对更多事情有模糊的概念。我们永远不会缺少需要澄清的想法。

The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification.

瑟斯顿将数学定义为以分享和理解为导向的人类合作项目，而不是对永恒真理的探索。没有人类的理解，定理就没有价值。谁在乎谁先证明了这个或那个结果？重要的是我们赋予这些结果的意义。真正的数学存在于我们每个人心中。

Thurston defines mathematics as a collaborative human project oriented toward sharing and understanding, not a search for eternal truths. Without human understanding, theorems have no value. Who cares who proved this or that result first? What counts is the meaning that we give to those results. Real math is the one that lives in each of us.

瑟斯顿的回应看似无伤大雅，但却是对两千多年来数学呈现方式的深刻质疑。这是本书的关键信息之一，我们稍后会再讨论它。

Thurston’s response might seem innocuous, but it’s a profound questioning of the way that math has been presented for over two millennia. It’s one of the key messages of this book and we’ll come back to it.

他继续说道：

He continues:

我们是具有深厚社会性和本能性的动物，我们的幸福取决于我们所做的许多难以用理智的方式解释的事情。

We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way.

光凭理性很可能会把你引入歧途。我们当中没有人足够聪明和智慧，能够理智地解决这一切。

Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

正是在这里，瑟斯顿使用“误入歧途”一词，插入了泰德·卡辛斯基 (Ted Kaczynski) 维基百科页面的链接，直接提及了卡辛斯基。

It’s at this precise point, with the word astray, that Thurston includes a direct reference to Ted Kaczynski by inserting a link to his Wikipedia page.

“光凭理性很可能会把你引入歧途”这句话听起来很平庸，是常识性的建议，但太过模糊和缺乏新意，不值得重视。事实并非如此：瑟斯顿是认真的。他很清楚自己在说什么，而且他的说法很准确。

Saying that “bare reason is likely to lead you astray” seems banal, commonsense advice but too vague and unoriginal to attach much importance to it. That’s not the case: Thurston is dead serious. He knows exactly what he’s talking about and he’s making a precise statement.

在第 14 章中，我们讨论了笛卡尔的计划，该计划依靠我们与生俱来的能力来发现真理，从而彻底重建整个科学和哲学。我们说，这种方法，即理性主义，遇到了笛卡尔未曾预见到的困难。

In chapter 14 we talked about Descartes’s project to reconstruct all of science and philosophy from the ground up, relying on our innate ability to find evident that which is true. We said that this approach, rationalism, encountered difficulties that Descartes hadn’t foreseen.

当瑟斯顿说“我们都不够聪明和智慧”以及“光凭理性很可能会让你误入歧途”时，他指的就是这个。

When Thurston says that “none of us are smart and wise enough” and that “bare reason is likely to lead you astray,” this is what he’s talking about.

数学思维为何如此强大？然而，它的极限是什么？理性的极限又是什么？我们如何区分数学推理和偏执狂之间有什么区别？这些问题的答案并不在于数学本身，而在于数学与我们的语言和智力内部运作之间的密切关系。

Why is mathematical thought so powerful? What, however, are its limits, and what are the limits of rationality? How do we distinguish between mathematical reasoning and paranoid delirium? The answer to these questions isn’t found in mathematics itself but in its close relationship with our language and the inner workings of our intelligence.

这个主题将会伴随我们读完本书。

This is the subject that will occupy us to the end of the book.

18

房间里的大象

18

The Elephant in the Room

你一直都知道理性存在问题。

You’ve always known there’s a problem with rationality.

它应该是我们文明的基础。无论如何，这就是他们在学校里告诉你的。我们被教导要以合乎逻辑和结构化的方式组织我们的想法。我们被教导要区分有效的推理和无效的推理。我们被教导要忽略那些不合逻辑、不严谨、不连贯的东西。

It’s supposed to be the basis of our civilization. At any rate that’s what they tell you in school. We’re taught to organize our ideas in a logical and structured manner. We’re taught to distinguish between a reasoning that’s valid and one that’s not. We’re taught to discount what isn’t logical, rigorous, coherent.

当然，没有人会傻到相信这个故事。我们只是假装相信。一旦课程结束，一旦学校大门关闭，我们就会继续过着我们的生活，好像这一切都不重要。

Of course, no one’s stupid enough to believe this story. We just pretend we do. Once the lesson is over, once the school door’s closed, we continue to live our lives as if all that was of no importance.

相信有一天我们能够变得完全理性，就如同相信有一天我们会停止吃甜食和油腻食物一样天真。

Believing that one day we can become entirely rational is as naïve as believing that one day we’ll stop eating sweet and fatty foods.

矛盾的是，它并不能阻止我们诉诸秘密理性。

The paradox is that it doesn’t stop us from having recourse to secret rationality.

当你真诚地专注于某件事时，当你遇到麻烦时，当你在工作中遇到问题或在家里遇到问题时，你会本能地求助于数学家使用的方法。

When you’re sincerely preoccupied with something, when you’re in trouble, when you have problems at work or problems at home, you instinctively call on the method used by mathematicians.

晚上，躺在床上，你试图理解这个问题。你反复思考。你在脑海中重现从记忆和想象深处挖掘出来的心理图像。你用这些图像玩乐高。你试图组织它们，将它们拼凑在一起，组装出一些有意义的、有意义的东西。

At night, in your bed, you try to understand the issue. You mull it over. You replay in your head mental images that you dig up from the depths of your memory and imagination. You play Lego with these images. You try to organize them, fit them together and assemble something meaningful, something that makes sense.

有时你会感觉你明白了一切。你的手下各种图像汇聚在一起。你用新的方式重新诠释了过去的事件。你注意到一个细节、一个新元素、一些你尚未察觉但就在你眼前的东西。

Sometimes you have the feeling you understand it all. Your mental images come together. You reinterpret a past event in a new way. You pick up on a detail, a new element, something that had been right under your nose that you hadn’t yet perceived.

现在你看到了，一切都说得通了。这是一个启示，一个让你兴奋并想与他人分享的发现。

Now that you see it, everything makes sense. It’s a revelation, a discovery that gets you excited and makes you want to share it with others.

你和你最好的朋友谈论这件事。但很快你就注意到她眼里有让你烦恼的东西。她似乎很生气。她很担心你。她能说的只有一句简单的话：“尽量不要找太多借口。”

You talk about it with your best friend. But soon enough you notice something in her eyes that troubles you. She seems annoyed. She’s worried about you. All that she can find to say is a simple phrase: “Try not to rationalize too much.”

最糟糕的是，你知道她是对的。当有人想出一个一切都太过完美合乎逻辑的推理时，你自己就会怀疑有什么不对劲。他想得太多了，这似乎有点可疑。

The worst thing is, you know she’s right. You yourself, when someone comes up with a reasoning where everything all fits together too neatly, suspect that something isn’t right. He’d thought about it too much, and it seems fishy.

例如，一个人在小木屋里呆了二十年，思考了我们文明中所有问题的根源，并提出了一份 232 个段落的宣言，其中所有内容都完美地衔接在一起——你不会觉得这完全令人放心。你不会告诉自己，“这个人一定是对的。”相反，你会说，“这个人不可能有很多朋友。”

For example, a guy who’s spent twenty years in a log cabin, thinking through the root cause of every problem with our civilization, and comes up with a manifesto of 232 numbered paragraphs where everything fits together all too well—you don’t find that entirely reassuring. You don’t tell yourself, “The guy must be right.” Instead you say, “The guy can’t have many friends.”

如果你对理性的不信任只是因为懒惰、不够努力，那也没什么大不了的。你可以让别人来做，并从他们的智慧中受益。

If your distrust of rationality was only a matter of intellectual laziness, of not trying hard enough, that would be no big deal. You could let others do the work and benefit from their wisdom.

问题在于你对理性的结果没有信心。你知道思考和推理并不总能揭示真相。有时你会有相反的印象：在某些情况下，理性会让你偏离真相。

The problem is that you don’t have confidence in the outputs of rationality. You know that thinking and reasoning don’t always uncover the truth. Sometimes you have the opposite impression: in some cases, rationality leads you astray from the truth.

这不是一个小问题。这是一个大问题。这是一个显而易见的问题——一个如此巨大、后果如此严重、对我们的生存如此重要、却从未被提及的问题。

It’s no small problem. It’s an enormous problem. It’s the elephant in the room—a problem so enormous and laden with consequence, so central to our existence, that it’s never spoken of.

如果人类想要给自己哪怕一丝机会来克服所面临的艰巨挑战，那么我们首先要确定笛卡尔的方法是否真的有效，这样不是更好吗？

If humanity wants to give itself the slightest chance to overcome the formidable challenges that it is facing, wouldn’t it be better if we could start by agreeing whether or not Descartes’s method actually works?

沙和泥

Sand and Mud

当笛卡尔制定从头开始重建科学和哲学的计划时，我们很容易看出他的出发点。

When Descartes formulated the project to reconstruct science and philosophy from the ground up, it’s easy to see where he was coming from.

他评论说，最伟大的学者无法就最基本的课题达成一致。他们所谓的知识往往只是“建立在沙子和泥土上的宏伟宫殿”。相反，数学是建立在坚固的岩石上的。这正是引起笛卡尔注意的地方：“我很惊讶，没有比这更崇高的东西建立在如此牢固和坚实的基础上。”

He remarked that the greatest scholars were incapable of agreeing on the most elementary subjects. Quite often their so-called knowledge consisted only of “magnificent palaces built on nothing more than sand and mud.” Mathematics, on the contrary, was built on solid rock. This is what caught Descartes’s eye: “I was astonished that nothing more exalted had been built on such sure and solid foundations.”

既然数学家使用的方法如此有效，既然它们产生的真理经久不衰，那么我们能否将它们应用于数学之外并产生不可动摇的真理呢？

Since the methods used by mathematicians are so effective, since they produce truths that survive millennia without showing their age, couldn’t we apply them outside of mathematics and produce unshakeable truths?

我们现在知道答案了，不幸的是，答案是不能。或者说，答案部分是肯定的，部分是否定的。

We now know the answer, and it is, unfortunately, no, we cannot. Or rather, the answer is partially positive and partially negative.

你可以在数学之外应用数学家使用的方法。如果我不想鼓励你这样做，我就不会写这本书。笛卡尔是对的，他的方法是理解世界的不可思议的工具，它确实可以让我们变得更聪明。无论如何，没有 B 计划。我们没有其他可以提供类似好处的替代方法。

You can apply the methods used by mathematicians outside of math. I wouldn’t have written this book if I didn’t want to encourage you to do so. Descartes was right, his method is an incredible tool for understanding the world, and it can literally make us more intelligent. At any rate, there is no plan B. We have no alternate method at our disposal that offers similar benefits.

但是当我们在数学之外使用它时，我们需要小心：只有在数学内部，这种方法才能产生不可动摇的真理。

But when we use it outside of mathematics, we need to be careful: it’s only within mathematics that this method is able to produce unshakeable truths.

这并不意味着你应该停止思考。谨慎并不意味着选择困惑和优柔寡断，并拒绝“自信地度过一生”。恰恰相反。合理化，寻找你不理解的事情的解释，是一件很棒的事情。为什么有人会选择保持无知呢？

That doesn’t mean you should stop thinking. Being careful doesn’t mean opting for confusion and indecisiveness, and refusing to “proceed with confidence through life.” Quite the contrary. Rationalizing, looking for explanations for things you don’t understand, is a great thing to do. Why would anyone choose to remain ignorant?

小心只是意味着记住笛卡尔的方法可以修改你的心理表象和直觉，并逐渐加强其内部一致性。

Being careful simply means keeping in mind that Descartes’s method has the effect of modifying your mental representations and your intuitions and gradually reinforcing their internal consistency.

这其实就是整个方法的重点。通过将我们的信念锚定在无可争辩的证据和严格的推理上，我们可以将它们变成确定性，随着时间的推移，这些确定性会变得像钢筋混凝土一样坚固。

It’s in fact the whole point of the approach. By anchoring our convictions in indisputable evidence and rigorous deduction, we can turn them into certitudes that, over time, become as strong as reinforced concrete.

但有时这些确定性是错误的。

Except that sometimes these certitudes are false.

一道牢不可破的墙

A Sort of Unbreachable Wall

你绝对肯定所有的鸡都是由其他鸡下的蛋孵化出来的。通过严密的推理，从逻辑的角度来看，你应该绝对肯定地推断出，从来没有第一只鸡，也没有第一颗蛋，鸡和蛋自古以来就存在了。因此，鸡和蛋在地球形成之前就存在了。

You’re absolutely certain that all chickens come from eggs, which are laid by other chickens. Through perfectly rigorous reasoning, from a logical perspective, you should thus deduce with absolute certainty that there was never a first chicken nor a first egg, that chickens and eggs have existed since the dawn of time. Chickens and eggs thus preexisted the formation of the planet.

这个例子很愚蠢，所以它很重要。如果如此简单的推理得出的结论有如此严重的缺陷，那么我们怎么能相信那些甚至不是愚蠢而是人为的、看似聪明的推理呢？

This example is idiotic and that’s why it’s so important. If such a simple reasoning leads to a conclusion so profoundly flawed, what confidence can we have in reasonings that aren’t even idiotic but, instead, are contrived and seem intelligent?

鸡和蛋之谜本应给我们带来一个“悖论”，一道人类理解无法逾越的墙，我们只能屈服于它。

The riddle of the chicken and the egg is supposed to present us with a “paradox,” a sort of unbreachable wall to human comprehension, before which we have no other choice but to bow down.

但悖论并不比诡计或本质上违反直觉的真理多。悖论总是暂时的状态，等待解决。将问题描述为结构上的悖论只是夸张地说你无法解决它。

But there aren’t any more paradoxes than there are tricks, or truths that are counterintuitive by nature. Being a paradox is always a temporary status, in wait of a resolution. Presenting a problem as structurally being a paradox is just a pompous way of saying you can’t solve it.

鸡和蛋之谜的表面解答呼唤进化论。我们看到的这只鸡的妈妈是另一只鸡，与第一只鸡略有不同。这第二只鸡的妈妈则更不同。到目前为止一切都很好。我们仍在谈论看起来像鸡和蛋的鸡和蛋。

A superficial resolution of the riddle of the chicken and the egg calls in the theory of evolution. The mother of the chicken we see is another chicken, slightly different from the first. The mother of this second chicken is a bit more different. Everything’s fine up to this point. We’re still talking about chickens and eggs that look like chickens and eggs.

但如果你回溯到 1.5 亿年前，你会遇到鸡妈妈的妈妈……鸡妈妈的妈妈看起来根本不像鸡，而是恐龙。再往前追溯，我们会遇到甚至不下蛋的动物。这也许不能告诉我们它们长什么样，但至少可以让我们确信，鸡在地球形成之前并不存在。

But if you go back 150 million years, you come across the mother of the mother . . . of the mother of the chicken, who doesn’t look like a chicken at all, but a dinosaur. Go back even further, and we come across animals that don’t even lay eggs. That might not tell us what they look like but at least it reassures us that chickens didn’t preexist the formation of the planet.

然而，这种解谜方式却忽略了最令人困惑的方面：为什么一开始会有这个谜？我们怎么会从一个无可争辩的正确假设出发，按照一个无可争辩的正确推理，得出一个无可争辩的错误结论呢？

This way of solving the riddle, however, leaves aside its most troubling aspect: why was there a riddle in the first place? How can it be that, starting from a hypothesis that is indisputably true, following a reasoning that is indisputably correct, we arrive at a conclusion that is indisputably false?

这是真正的先有鸡还是先有蛋的谜题，其答案已为人所知近一个世纪。但这个答案是如此令人震惊、不安和影响深远，以至于它被学校刻意掩盖和忽略。

That’s the true riddle of the chicken and the egg, and its solution has been known for almost a century. But this solution is so staggering, unsettling, and consequential that it’s carefully obscured and left out of schools.

谜题之所以存在，是因为它的结构与逻辑推理不相容，我们永远无法100%确定用人类语言表达的、通过演绎逻辑得出的真理。

Here is the reason why there was a riddle: human language is structurally incompatible with logical reasoning, and we can never have 100 percent certainty in truths expressed in human language and arrived at through deductive logic.

这适用于各种“真理”，无论是来自官方科学的真理，还是来自我们日常使用的琐碎推理的真理。无论我们是否表达自己的观点，我们是使用文字推理，遵循结构化且逻辑连贯的论点，还是秘密操纵头脑中的心理图像。我们是否有意识地思考某件事，还是让自己受直觉的引导，这都无关紧要。

That goes for all kinds of “truths,” the ones coming from official science and the ones coming from our small everyday reasoning that we employ all the time. And it doesn’t matter whether we express our reasoning in words, following structured and logically coherent arguments, or whether we secretly manipulate mental images in our head. It doesn’t matter whether we’re aware of thinking something through, or whether we let ourselves be guided by our intuition.

可悲的现实是，与笛卡尔所说的相反，“我们非常清楚、明确地设想的事物”并不总是正确的。

The sad reality is that contrary to what Descartes claimed, the “things that we conceive of very clearly and distinctly” are not always true.

笛卡尔忽略了一个基本点：所有的推理，即使是最坚实的推理，随着远离日常经验，最终也会变得支离破碎，这并不是因为缺乏严谨性，而是因为我们的语言本身就是建立在沙泥的基础上的。

Descartes missed an essential point: all reasoning, even the most solid, ends up coming apart the further it gets from day-to-day experience, not for lack of rigor but because our language itself is built on a base of sand and mud.

唯一的例外是数学推理，当它以数学的官方语言表达时。如果这种人造语言如此不人性化，与我们通常的思维方式如此不相容，原因很简单：它倾向于与逻辑推理相容。

The only exception is mathematical reasoning when it is articulated in the official language of mathematics. If this artificial language is so inhuman and so incompatible with our usual way of thinking, it’s for a very simple reason: its bias is to be compatible with logical reasoning.

当我们想远离日常具体经验时，逻辑形式主义会指引我们。这是我们唯一可以自由发挥理性冲动的工具，没有限制，没有情结或禁忌。

When we want to venture far from our everyday concrete experience, logical formalism helps guide us. It’s the only tool at our disposal that lets us give free rein to our impulse toward rationality without limits, without complexes or taboos.

在数学之外，理性仍然受到我们语言和我们感知世界的方式的脆弱性的不断威胁。

Outside of mathematics, rationality remains under constant threat from the fragility of our language and our way of perceiving the world.

大象不存在

Elephants Don’t Exist

在第六章中，我们已经指出数学的官方语言——逻辑形式主义，对人类来说是一种外语，并且我们举了这个例子来说明它是如何运作的：在数学中，如果有鼻子是大象的定义的一部分，那么没有鼻子的大象就不再是大象了。

In chapter 6, we already stated that the official language of mathematics, logical formalism, is a foreign language for humans, and we gave this example to show how it works: in mathematics, if having a trunk is part of the definition of an elephant, then an elephant without a trunk immediately ceases being an elephant.

这种与词语含义的奇怪而僵化的关系是数学方法的核心。这是一个经常让初学者感到震惊的奇怪现象。但对此有一个简单的解释：没有这种刚性，任何语言都无法与逻辑推理兼容。这可能很烦人，但事实就是如此。

This bizarre, perversely rigid relationship to the meaning of words is at the heart of the mathematical approach. It’s an oddity that often shocks beginners. But there’s a simple explanation for it: without this rigidity, no language can be compatible with logical reasoning. It may be annoying, but that’s how it is.

任何利用大象有鼻子这一事实进行的有关大象的推理，都会因发现一头没有鼻子的大象而失效。

Any reasoning about elephants that makes use of the fact that they have trunks will be invalidated by the discovery of a single elephant without a trunk.

在第 8 章中，我们在讨论字典的缺点时触及了一个相关主题。从远处看，字典看起来就像是真的一样。我们总是希望自己使用的单词有明确的定义，我们说的短语有准确的含义。但一旦你深入挖掘，就会发现循环定义。

In chapter 8, we touched upon a related topic when discussing the shortcomings of dictionaries. At a distance, dictionaries look like the real thing. We always want to believe that the words we use are solidly defined and that the phrases we speak have a precise meaning. But once you scratch a bit below the surface, you find circular definitions.

错误在于相信编纂词典的人没有正确地履行他们的职责，而相信有更好、更聪明、更严格的方法来定义我们使用的词汇。

The error would be in believing that people who put together dictionaries aren’t doing their job correctly, and that there’s a better, smarter, and more rigorous way of defining the words we use.

但是，字典定义如此不充分有着一个深层的结构性原因：在我们的语言中，我们不可能真正地定义单词，而且我们与世界的关系远没有我们想象的那么牢固。

But there’s a deep structural reason why dictionary definitions are so deficient: it’s rigorously impossible to truly define words in our language, and our relationship to the world is much less solid than we would like to believe.

从日常语言中的一个单词开始，尝试巩固其定义，并发现这是一项不可能完成的任务：这是一个令人不安但又有教育意义的经历，你应该在一生中至少尝试一次。

Starting with a word from everyday language, trying to solidify the definition, and finding out that it’s an impossible task: it’s a troubling yet instructive experience that you should take the time to try at least once in your life.

让我们用大象来做这件事。我在一本字典里找到了这个定义：“一种体型粗壮、通常体型极大、几乎没有毛发的食草哺乳动物，鼻子细长，形成肌肉发达的躯干，上颌有两颗门牙，雄性的上颌会长出长长的象牙。”

Let’s do that with elephants. I found this definition in a dictionary: “a thickset, usually extremely large, nearly hairless, herbivorous mammal that has a snout elongated into a muscular trunk and two incisors in the upper jaw developed especially in the male into long ivory tusks.”

这种方法的优点是实用。它列出了大象可能具有的不同特征。然而，这是一个循环定义：你认为他们如何定义“象鼻”？或“象牙”？或“象牙”？

The approach has the merit of being pragmatic. It consists of listing the different characteristics you can expect to find in an elephant. It is nevertheless a circular definition: how do you think they define a “trunk”? or a “tusk”? or “ivory”?

除了循环性之外，这个定义还有一个缺陷，那就是它没有提及一个基本方面，那就是动物物种的概念。

Apart from circularity, this definition also has the flaw of being silent on an essential aspect, the concept of animal species.

显而易见，我们面对的不是孤立的个体动物，而是可以归类为物种的生物。如果我们发明了大象这个词，那是因为我们觉得眼前的大象都有一些共同点。我们所说的“大象”就是它们之间的共同点，即物种。

It’s obvious to all that we’re not dealing with animals that are isolated individuals, but with beings that we can group into species. If we’ve invented this word elephant, it’s because we had the impression that the individual elephants in front of our eyes all shared something in common. By “elephant,” it’s this common thing between them that we’re referring to, the species.

事实上，如大家所知，大象不只分为一个物种，而是两个物种：非洲象和亚洲象。

In reality, as you know, elephants make up not one but two species, African elephants and Asian elephants.

事实上，情况比这更复杂。二十多年来，生物学家们已经知道非洲象有两种不同的物种：草原象，学名为Loxodonta africana，以及森林象，学名为Loxodonta cyclotis。亚洲象属于第三种，学名为Elephas maximus。

In fact, it’s more complicated than that. For more than twenty years biologists have known that there are two distinct species of African elephants: the savanna elephant, whose scientific name is Loxodonta africana, and the forest elephant, Loxodonta cyclotis. Asian elephants form a third species, Elephas maximus.

如果你真的想对大象这个词给出一个非循环的、非常科学的定义，你可能会得出这样的结论：“对三个物种的代表给予的通用名称，即非洲象、环象和大象。 ”

If you really wanted to give a noncircular and seriously scientific definition of the word elephant, you might come up with something like “Generic name given to representatives of three species, Loxodonta africana, Loxodonta cyclotis, and Elephas maximus.”

但现在你必须对非洲象、环象和大象给出一个严肃的定义。尽管这看起来令人震惊，但没有人能够做到这一点。

Except that now you’d have to give a serious definition of Loxodonta africana, Loxodonta cyclotis, and Elephas maximus. And as shocking as it may seem, no one is able to do so.

实际上，生物学家定义物种是从某个特定个体开始的，这个个体被称为正模标本，作为参考点。例如，有一头特定的大象（已经死了很久）荣幸地成为非洲象的正模标本。它是“零号大象”，是该物种的旗手。从科学的角度来看，非洲象只不过是一群与这头零号大象“属于同一物种”的个体，无论是活着的还是死去的。

In practice, biologists define species as starting from a given individual, called a holotype, that serves as a point of reference. For example, there’s a specific elephant (who’s been dead for a long time) that has the honor of serving as the holotype for Loxodonta africana. It is “elephant zero,” the standard-bearer of the species. From a scientific point of view, Loxodonta africana is nothing other than the group of individuals, living or dead, “of the same species as” this elephant zero.

剩下的就是为“同种”赋予一个准确的含义。这就是事情变得棘手的地方。

It simply remains to give a precise meaning to “of the same species as.” And here’s where things get tricky.

在任何合理的动物物种定义中，你都希望能够说母亲与其子女属于同一物种。但是，如果你一方面选择大象母亲的母亲……0 号大象母亲的母亲，另一方面选择你自己母亲的母亲……在某个时刻，你会得到同一种雌性，它们属于一种灭绝的哺乳动物，这种哺乳动物与恐龙一起生活了 1.5 亿多年。这和先有鸡还是先有蛋的问题一模一样。合乎逻辑的结论是，你是一头大象。

In any reasonable definition of an animal species, you’d want to be able to say that a mother is of the same species as its children. But if you take on one hand the mother of the mother . . . of the mother of elephant zero, and on the other hand the mother of the mother . . . of your own mother, at a certain point you’ll get to the same female, belonging to a species of extinct mammals that lived more than 150 million years ago, alongside dinosaurs. It’s the exact same problem of the chicken and the egg. The logical conclusion is that you are an elephant.

为了避免这个问题，有必要设定一个限制：你和你的母亲是同一个物种，和你母亲的母亲是同一个物种，等等，但你不能无限期地继续下去。

To avoid this problem, it’s necessary to set a limit: you’re of the same species as your mother, and the mother of your mother, and so on, but you can’t continue that indefinitely.

如何确定这个极限？官方的解决方案依赖于相互生育的概念，并得出以下定义，你可以在维基百科的“物种”下找到它：“在生物学中，物种通常被定义为最大的生物群体，其中任何两个具有适当性别或交配类型的个体都可以产生可育后代，通常是通过有性生殖。”

How do you determine the limit? The official solution relies on the concept of interfertility and leads to the following definition, which you can find on Wikipedia under “species”: “In biology, a species is often defined as the largest group of organisms in which any two individuals of the appropriate sexes or mating types can produce fertile offspring, typically by sexual reproduction.”

生育后代的问题主要涉及驴和马，它们可以产下不育的后代（骡子）。因此，它们是两个不同的物种。

The issue of fertile offspring notably concerns donkeys and horses, which can produce offspring (mules) that are, however, sterile. They are thus two different species.

你可能认为我们终于搞定了。但事实远非如此。根据这个定义，不育个体本身就是一个物种。当你给猫做绝育手术时，它就变成了另一个物种。这显然毫无意义，但这是严格执行这个定义的结果。

You might think that we’ve finally nailed it. But that’s far from being the case. According to this definition, a sterile individual is a species unto itself. When you neuter your cat, it changes species. That obviously makes no sense, but it’s the result of the strict application of the definition.

更严重的是，如果后代通常但不总是不育，你会怎么做？有许多有记录的可育骡子的例子。你在哪里设定界限？是否有必要固定一个阈值，武断地规定产生可育后代的概率小于 1％，就意味着你面对的是两个不同的物种？

More seriously, what do you do when the offspring are generally but not always sterile? There are a number of documented examples of fertile mules. Where do you set the limit? Is it necessary to fix a threshold, arbitrarily decree that less than 1 percent of a chance of producing fertile offspring means you’re dealing with two distinct species?

相互生育的概念本质上是模糊且成问题的。当两个物种分离时，正是定义最有价值的时刻，它却停止发挥作用。这涉及到我们自己物种的起源：我们的 DNA 带有与尼安德特人杂交的可育痕迹。这是否意味着智人和尼安德特人属于同一个物种，这与大家普遍承认的相反？

The notion of interfertility is intrinsically vague and problematic. When two species separate, at the precise moment when the definition would be of most value, it ceases to function. This concerns the origins of our own species: our DNA carries traces of fertile hybridizations with Neanderthals. Does this mean that Homo sapiens and Homo neanderthalensis form one and the same species, contrary to what is communally admitted?

除了这些理论上的不一致之外，生物物种概念还带来了巨大的实际问题。如果我想知道我是不是一头草原象，我应该与一头雌性非洲象交配。如果她没心情交配我该怎么办？或者如果她因为时机不对而没有怀孕怎么办？我需要尝试多少次？

Apart from these theoretic inconsistencies, the biological species concept creates immense practical issues. If I want to know whether I’m a savanna elephant, I should mate with a female Loxodonta africana. What do I do if she’s not in the mood? Or if she doesn’t get pregnant because it’s not the right moment? How many times do I need to try?

生物学家应该比数学家有更发达的实践意识。我很想知道他们是如何做到的。

Biologists are supposed to have a more developed practical sense than mathematicians. I’d be curious to know how they do it.

模糊地了解我们的意思

Vaguely Knowing What We Mean

所有这一切中最奇怪的是，我们无法对大象给出 100% 严格的定义，而这一概念却在我们的直觉中呈现出来，这两者之间形成了鲜明的对比。

The strangest thing about all this is the contrast between our inability to give a 100 percent rigorous definition of an elephant, and the evidence with which the concept presents itself to our intuition.

当我们听到“大象”这个词时，我们会觉得我们非常清楚它的含义。当我们遇到大象时，我们马上就能认出它。我们对大象的概念非常清晰。

When we hear the word elephant, we have the impression that we know very well what it means. When we encounter an elephant, we recognize it right away. We have a perfectly clear notion of what elephants are.

只有当我们试图明确我们得出的第一个定义周围的轻微模糊性时，当我们想使它与逻辑推理兼容时，问题才会出现。我们的尝试我们总是以不连贯和不一致的方式思考，而每次我们都必须用新的科学发现来解决这些不一致，而这又会导致新的不一致。

The problems come along only once we try to specify the slight vagueness that surrounds the first definition we come up with, when we want to make it compatible with logical reasoning. Our attempts to specify our thinking always end in incoherence, creating inconsistencies that we have to resolve each time with new scientific discoveries, which in turn lead to new inconsistencies.

一些简单、具体且显而易见的事情似乎无法用语言来表达。

Something simple, concrete, and plainly obvious seems impossible to really capture in words.

这种怪异现象并非大象所特有。这是一种普遍现象，反映了我们心理过程的神经基础。我们将在下一章中回顾这一点。

This oddity isn’t specific to the notion of an elephant. It’s a universal phenomenon that reflects the neurological underpinnings of our mental processes. We’ll come back to this in the next chapter.

我们无法准确定义词语的含义，最引人注目的例子来自查尔斯·达尔文本人 1859 年在《物种起源》第二章的开篇： “我也不打算在这里讨论物种一词的各种定义。迄今为止，还没有一个定义能让所有博物学家都满意；然而，每个博物学家在谈到物种时，都模糊地知道自己的意思。”

The most spectacular illustration of our inability to give a precise meaning to words comes from the pen of Charles Darwin himself, in 1859, in the opening lines of the second chapter of Origin of Species: “Nor shall I here discuss the various definitions which have been given of the term species. No one definition has as yet satisfied all naturalists; yet every naturalist knows vaguely what he means when he speaks of a species.”

在一部永恒的科学杰作中发现如此公然承认其弱点，这很能说明问题。这个问题根深蒂固，我们对此无能为力。

It is quite telling to find such a blatant admission of weakness at the heart of a timeless scientific masterpiece. The issue runs deep and there’s not much we can do about it.

达尔文对“物种”一词的含义知之甚少，但他却写了一整本书来介绍它。

Darwin knew only vaguely what he meant by species, and yet he wrote an entire book about it.

两种语言，两套规则

Two Languages, Two Sets of Rules

最后，如果你真的想了解什么是数学以及为什么它在科学中无处不在，你必须从我们语言的脆弱性开始。

In the end, if you really want to understand what math is and why it is omnipresent in science, you have to start with this fragility of our language.

数学的历史与我们对理性的渴望一样悠久，为此，数学旨在确保和稳定词语的含义。将数学限制在对数字和形状的研究是一种常见的错误。除了技术方面、定理和方程式之外，数学首先是一种使用语言和赋予词语意义的不同方式。

Mathematics is as old as our desire for reason, and, to this end, to secure and stabilize the meaning of words. It’s a common error to restrict it to the study of numbers and shapes. Beyond its technical aspects, beyond theorems and equations, math is above all a different way of using language and attributing meaning to words.

人类语言和数学语言在数千年间并行发展。如今，它们已经如此紧密地交织在一起，以至于很难区分彼此。在日常生活中，在最普通的对话中，我们通常会在两种使用词语的方式之间来回切换，但通常我们并没有意识到这一点。

Human language and mathematical language have evolved in parallel over millennia. Today they have become so intertwined that it’s become hard to tell one from the other. In day-to-day life, in the most ordinary kinds of conversations, we navigate between these two ways of using words, generally without being aware of it.

这适用于我们所有人，即使是那些认为自己讨厌数学的人。无论我们的背景和学术水平如何，我们都对数学方法有一定的了解，并且每天都会用到它所支持的思维方式。

That goes for all of us, even people who think they hate math. Whatever our background and academic level, we have all acquired a certain degree of familiarity with the mathematical approach and have daily recourse to the modes of thinking it enables.

而且由于没有人向我们解释过整个过程是如何运作的，我们总是被绊倒。我们从一种语言转到另一种语言，却忘记了它们遵循两种完全不同的逻辑。在某些情况下，这会导致严重的后果。

And since no one’s ever explained to us how the whole thing works, we keep getting tripped up. We pass from one language to the next, forgetting that they follow two totally different logics. In some cases, this leads to serious consequences.

每一种语言都有自己的功能，自己的一套有自己的规则，有自己的优势，也有自己的劣势，但这些都同样不可或缺。

Each of these languages has its own functions, its own set of rules, its own strengths, its own weaknesses. And they’re equally indispensable.

两种语言，两套规则

Two languages, two sets of rules

相信字面意思

Taking Words at Face Value

两种语言经常使用相同的词语。改变的是我们如何赋予这些词语意义。

The two languages often use the same words. What changes is how we attribute meaning to those words.

球体这个词就是一个很好的例子。当你听到有人说“地球是球形的”时，你很清楚这句话的意思。

The word sphere is a good example. When you hear someone say, “The Earth is shaped like a sphere,” the meaning of these words is pretty clear to you.

如果您和我一样，认为该说法正确，那是因为您在人类语言中对“领域”一词的解释是感性的、模糊的。

If, like me, you think the statement is correct, it is because you interpret the word sphere in human language, in a perceptual and vague manner.

然而，用数学语言来说，这个说法显然是错误的：球体上不可能有山。

In mathematical language, however, the statement is clearly false: spheres can’t have mountains.

在数学中，词语的定义是“公理化的”：通过形式定义来完全描述它们。它们是虚构的、完美的、固定的结构：球体是“三维空间中与中心等距的所有点的集合”。你无法改变这一切。如果你取出一个点，甚至稍微移动一个点，它就不再是球体了。

In mathematics, words are defined “axiomatically”: via formal definitions that characterize them entirely. They are imaginary, perfect, and fixed constructions: a sphere is “the set of all points in three-dimensional space that are located at an equal distance from a center.” You can’t change any of this. If you take out or even slightly shift a single point, it ceases to be a sphere.

最糟糕的是，“球体”在人类语言词典中的定义完全相同。唯一改变的是我们与定义的关系。

The worst thing is that “sphere” has the exact same definition in human-language dictionaries. The only thing that changes is our relationship to the definition.

在人类语言中，没有人会真正地从字面上理解单词的含义。在现实生活中，你所谓的球体对应于你感知到的球体。要判定给定形状是球体，你必须有一个公差范围。橙子是球体，苹果或多或少是球体，梨不是球体。我挑战你写下你对球体的感知定义中隐含的公差范围的确切轮廓。

In human language, no one ever really takes words at face value. In real life, what you call a sphere corresponds to what you perceive of as a sphere. To decree that a given shape is a sphere, you have a tolerance margin. An orange is a sphere, an apple is more or less a sphere, a pear isn’t a sphere. I challenge you to write down the exact contours of the tolerance margin that is implicit in your perceptual definition of a sphere.

理性主义与经验主义

Rationalism vs. Empiricism

更有趣的是，你可以用另一种方式来做。你可以从人类语言中的一个单词开始，假装把它当作数学语言中的一个单词。这就是我们推理时所做的。

More interestingly, you can do it the other way. You can start with a word in human language and pretend to treat it as a word in mathematical language. It’s what we all do whenever we reason.

这有点棘手，但我们在不知不觉中就做到了这一点。我们从人类语言开始，转到数学语言进行推理，然后回到人类语言。每次我们提出假设并试图从中得出结论时，我们都会这样做。

It’s a bit tricky, but we’re experts at doing it without ever noticing. We start with human language, shift to mathematical language for reasoning, and return to human language. We do this each time we formulate hypotheses and try to draw conclusions from them.

这种日常活动是所谓科学方法的一个例子。我将通过一个相当简单但忠实地说明该过程中所有步骤的例子来总结如下。

This day-to-day activity is an instance of what is pompously called the scientific approach. I’ll sum it up as follows through an example that is rather simplistic but faithfully illustrates all the steps in the process.

比如，如果你声明，根据定义，大象是“非洲象、环象和大象三个物种之一的代表”，如果你相信这个定义陈述了绝对真理，那么就可以从中得出合乎逻辑的结论。

If you state, for example, that by definition an elephant is “a representative of one of three species, Loxodonta africana, Loxodonta cyclotis, and Elephas maximus,” if you make believe that this definition states an absolute truth, then it is possible to draw logical conclusions from it.

这就是字典定义的用途：它们作为暂时固定词语含义的数学模型，让我们有机会用它们进行推理。

That’s what dictionary definitions are for: they serve as mathematical models that temporarily anchor the meaning of words, giving us a chance to reason with them.

当你面前有一头既不是非洲象也不是大象的大象时，你可以推断它是环斑象。在你的模型中，这个推断是 100% 可靠的。你绝对确定：你“算过了”。

When you have an elephant in front of you that is neither Loxodonta africana nor Elephas maximus, you can deduce that it’s Loxodonta cyclotis. Within your model, this deduction is 100 percent reliable. You’re absolutely sure: you “did the math.”

但这种推论在现实生活中正确吗？这完全取决于你的模型的可靠性。也许你运气不好（或者非常幸运），你面前有第四种大象，这种大象非常罕见，至今还没有被描述过。

But is this deduction correct in real life? It all depends on the reliability of your model. Perhaps you’re out of luck (or incredibly lucky) and you have a fourth species of elephant before you, so rare that it hasn’t yet been described.

在人类语言中，没有什么是 100% 可靠的。我们总是感到惊讶。这就是为什么科学理论只能做出预测，然后可以通过经验来验证（以及理论获得了可信度）或被证明是错误的（在这种情况下你需要更改模型）。

In human language, nothing is ever 100 percent reliable. We’re constantly being surprised. It’s why a scientific theory only ever makes predictions, which can then be validated by experience (and the theory gains credibility) or disproven (in which case you need to change the model).

模型本身没有好坏之分。只要你不试图从中推断出山脉不存在，那么说地球是球形的就是一个很好的模型。我们技术的成功证明了科学方法有效：即使它不能产生绝对真理，科学也为我们提供了一种强大的思维方法，其预测与现实足够接近，具有实际用途。

Models are not good or bad in and of themselves. Saying that the Earth is shaped like a sphere is a good enough model as long as you don’t try to deduce from it that mountains don’t exist. The success of our technology is proof that the scientific approach works: even if it doesn’t produce absolute truths, science gives us a powerful method of thinking and its predictions are close enough to reality to be of practical use.

最终，理性应该被用作指导，而不是最终的裁判。我们眼前的现实总是比我们头脑中的确定性更值得关注。理性是伟大的，但经验主义者确实有道理。

In the end, rationality should be used as a guide rather than an ultimate judge. The reality that’s before our eyes always merits more attention than the certitudes in our heads. Rationality is great, but empiricists do have a point.

过于相信理性，使用人类语言时，好像它具有数学语言的所有属性，好像单词具有精确的含义，好像每个细节都值得解释，论点的逻辑有效性足以保证其结论的有效性，这是偏执狂的典型症状。当数学推理应用于数学之外且没有任何保障时，它就变成了一种真正的疾病。

Trusting reason too much, using human language as if it had all the attributes of mathematical language, as if words had a precise meaning, as if each detail merited being interpreted and the logical validity of an argument sufficed to guarantee the validity of its conclusions, is a characteristic symptom of paranoia. When applied outside of mathematics and without any safeguards, mathematical reasoning becomes an actual illness.

撕裂的蜘蛛网

A Torn Spider’s Web

我们祖先对过度思考的沉迷无疑是围绕真理概念的严重误解的根源。

Our ancestral addiction to overthinking is without a doubt the source of terrible misunderstandings that surround the notion of truth.

当我谈到“真理”时，我指的是数学家的真理，绝对的、永恒的真理，有些人喜欢把它写成真理，或真理，有时甚至是真理。

When I speak of “truth,” I’m speaking of the truth of mathematicians, the absolute and eternal truth, what some people like to write as Truth, or TRUTH, or sometimes even TRUTH.

这种真理是一个数学概念。它的存在方式与数字 5 或三角形和矩形相同。毫无疑问，它是这是数学史上的第一个发明，它先于其他所有发明，并对我们的文化产生了最大的影响。

This sort of truth is a mathematical concept. It exists in the same way as the number 5 or triangles and rectangles. It is undoubtedly the first invention in the history of mathematics, that which preceded all the others and which has had the greatest impact on our culture.

数学上的“真理”在人类语言中也有对应词，就像“球体”一样。但人类的版本非常成问题。就像不耐运输的水果一样，真理的概念在翻译中会受到影响。受损的球体看起来仍然像球体，但受损的真理看起来什么都不像。

Mathematical “truth” has its counterpart in human language, just like “sphere.” But the human version is very problematic. Like a fruit that doesn’t tolerate shipping well, the concept of truth suffers in translation. A damaged sphere still looks like a sphere, but a damaged truth doesn’t look like anything at all.

此外，我们从不期望人类语言中的陈述是明确且不可动摇的“真实”。我们只是期望它们清晰、富有表现力、有趣、诚实、真诚，并且能够教会我们一些关于世界的有用和相关的东西。

Besides, we never expect statements in human language to be definitively and implacably “true.” We simply expect them to be clear, expressive, interesting, honest, sincere, and able to teach us something useful and relevant about the world.

当我们说某事是“真实的”时，我们的意思绝不是字面意义上的。我们只是用这个词来表示所有其他事物，因为否则我们就没有机会使用它。

When we say that something is “true,” we never mean it literally. We only ever use the word as a shortcut for all these other things, because otherwise we’d have no occasion to use it.

这种情况当然令人沮丧。我们希望世界更加清晰、更加稳定。我们希望真理更加坚实，更少地依赖于我们的观点。

This situation is of course frustrating. We’d like the world to be clearer and more stable. We’d like truth to be more solid and less dependent on our point of view.

奥地利哲学家路德维希·维特根斯坦 (1889-1951) 完美地总结了这种挫败感：“我们越仔细地审视实际语言，它与我们的要求之间的冲突就越大。”

The Austrian philosopher Ludwig Wittgenstein (1889–1951) perfectly summed up this frustration: “The more closely we examine actual language, the greater becomes the conflict between it and our requirement.”

逻辑只有在词语的定义明确、完全精确且长期稳定的情况下才能发挥作用。尽管我们付出了巨大的努力，但我们仍无法在数学之外得出此类定义。维特根斯坦断言，这是一项不切实际的追求：“我们感觉就像必须用手指修补一张破损的蜘蛛网。”

Logic doesn’t function unless words have a definition that is explicit, perfectly precise, and stable over time. Despite immense efforts, we’ve been unable to produce these kinds of definitions outside of mathematics. Wittgenstein affirms that it’s a quixotic quest: “We feel as if we had to repair a torn spider’s web with our fingers.”

维特根斯坦承认了我们语言的内在局限性，这是 20 世纪最伟大的哲学突破之一。这使他打破了数千年来形而上学主导的传统，在形而上学中，哲学家们认为，利用理性可以解决那些难以解决的问题。这与先有鸡还是先有蛋的问题惊人地相似：这些问题与我们的日常生活相距甚远，它们的发生只是因为我们的语言失去了控制。

In acknowledging the intrinsic limitations of our language, Wittgenstein made one of the great philosophical breakthroughs of the twentieth century. This allowed him to break with a multi-millennial tradition dominated by metaphysics, in which philosophers believed that it was possible to attack, using rationality, problems that were strikingly similar to that of the chicken and the egg: problems so remote from our daily experience that they were occurring only as a result of our language losing its grip.

令人惊讶的是，维特根斯坦的哲学在专业圈子之外并不为人所知，因为它蕴含着非常实用的人生教训：我们应该接受循序渐进，在前进的过程中澄清我们的语言，并经常感到惊讶。这是治疗偏执症的有效方法。

It’s surprising that Wittgenstein’s philosophy isn’t more well known outside of specialized circles, as it holds a very practical life lesson: we should accept going step by step, clarifying our language as we move forward, and being regularly surprised. It’s an effective antidote for paranoia.

我们应该从他身上学到的另一个重要教训是有关我们教授数学的方式。这一教训与瑟斯顿的言论非常吻合：“数学的产物是清晰和理解。而不是定理本身。”

Another key lesson we should have drawn from him concerns the way we teach mathematics. The lesson is very much in line with Thurston’s remark that “The product of mathematics is clarity and understanding. Not theorems, by themselves.”

如果数学只能产生永恒的真理，那它就毫无用处，因为在人类经验中没有永恒真理的空间（因为我们的语言根本不允许）。

If math were good only for producing eternal truths, it would be of strictly no use, since there is no place for eternal truths in the human experience (as our language simply doesn’t permit it).

然而，我们仍在继续教授数学。这表明我们仍然相信数学在某些方面是有用的。那么数学到底有什么用呢？

Yet we continue to teach math. This indicates that we continue to believe that math is useful in a certain way. So what is it that math is good for?

为了找出答案，了解数学究竟如何运作以及它能为我们做些什么，我们不能继续忽视它最直接的实际方面：数学作用于我们的大脑并改变我们看待世界的方式。

To find out, to understand how math really works and what it can really do for us, we cannot continue to overlook its most direct practical aspect: math works on our brain and modifies how we see the world.

19

抽象和模糊

19

Abstract and Vague

你熟悉这种视觉错觉吗？它是最古老、最著名的视觉错觉之一。

Are you familiar with this optical illusion? It’s one of the oldest and most famous.

你看到了什么？

What do you see?

仔细看看。

Look closely.

大多数人会说他们看到了大象。我想你也看到了。但你能看到别的东西吗？

Most people will say they see an elephant. I imagine you see one as well. But can you see something else?

慢慢来。

Take your time.

在翻开这一页阅读下面的内容之前，给自己一个真正的机会。

Give yourself a real chance before turning the page and reading what follows.

这种幻觉最特别之处在于，要消除它需要花费很大力气。几乎不可能看不到大象。然而，如果你仔细观察，你会发现根本没有大象，只有纸上的墨水。

The most extraordinary thing about this illusion is that it takes quite a bit of effort to make it go away. It’s nearly impossible not to see an elephant. And yet, if you look closely, you’ll notice that there is no elephant, just ink on a page.

纸张上的墨水和大象之间的区别无疑是巨大的。是什么神秘现象让我们在只有墨水的纸张上看到了大象？

Between ink on page and an elephant, the difference is indisputably massive. What mysterious phenomenon makes us see an elephant where there’s only ink on a page?

这种视觉错觉如今被称为绘画。当然，绘画远不止是一种错觉。绘画具有风格和艺术价值，它传达着一种信息、一种象征和文化意义，而这些意义不能被简化为其具象内容。

This type of optical illusion is nowadays called a drawing. A drawing, of course, is a lot more than an illusion. A drawing has style and artistic value, it carries a message, a symbolic and cultural significance that can’t be reduced to its figurative content.

然而，如果我们能立即认出洞穴墙壁上画的猛犸象，而我们对旧石器时代的人们的语言、习俗和信仰一无所知，那是因为图画中的某些东西超越了任何文化惯例。即使我们没有密码，我们也能理解它。我们的大脑会自动将画中的动物与真实的动物联系起来。正是从这个意义上说，图画才是真正的视觉错觉。

And yet, if we instantly recognize mammoths painted on cave walls by Paleolithic people whose language, customs, and beliefs we know nothing of, it’s because something in the drawing goes beyond any cultural convention. Even if we don’t have the codes, we understand it all the same. Our brain automatically makes the connection between a drawn animal and a real animal. It’s in this way that drawings are true optical illusions.

对图画的视觉理解从婴儿时期就开始发展，不需要任何特殊的教导。婴儿在会说话之前就能理解图画。他们对图画的理解非常透彻，以至于我们依靠图画来教授词汇。如果图画只是文化习俗，那么情况就正好相反：婴儿会从学习词汇开始，然后学会如何识别图画。

Visual comprehension of drawings develops from infancy and doesn’t require any special teaching. Babies understand drawings before being able to talk. They understand them so well that we rely on drawings to teach vocabulary. If drawings were only cultural conventions, it would be the other way around: babies would begin by learning vocabulary and then be able to learn how to recognize drawings.

已知许多动物物种（不仅仅是哺乳动物）无需经过训练就能识别绘画对象。理解绘画并非人类的特权。

A number of animal species (and not just mammals) are known to be capable of recognizing drawn objects without having been taught. Understanding drawings isn’t a strictly human privilege.

绘画中产生的幻觉确实非常奇妙。

The illusion at work in a drawing is something really extraordinary.

想想看，你从本章开头的小草图中收集了多少信息。大象幼年还是年老？危险还是无害？它生气吗？它自信吗？你感到同情还是不信任？

Just consider the amount of information that you managed to gather from the little sketch that opened this chapter. Is the elephant young or old? Dangerous or harmless? Is it angry? Does it have a lot of self-confidence? Do you feel sympathy or distrust?

你从未上过解读图画的课程，但你知道如何立即、毫不费力地回答这些问题。所有这一切都来自在纸上画几条线。

You’ve never taken a class on interpreting drawings and yet you know how to answer these questions, immediately and without any effort. All that from a few lines drawn on a page.

从生物学角度看，这种奇迹怎么可能实现？

How is such a miracle even biologically possible?

当前科学能够提供的解释将有助于阐明我们的学习过程的本质，以及本书一开始所描述的心理可塑性实际上是如何运作的。

The explanation that current science is able to provide will help clarify the nature of our learning process, and how the mental plasticity described since the beginning of this book actually operates.

视觉的奥秘

The Mystery of Vision

视觉是一种复杂的现象，涉及光学、生物化学和神经学。视觉器官不仅包括眼睛，还包括视神经，最重要的是大脑。

Vision is a complex phenomenon that concerns optics, biochemistry, and neurology. The organs of vision comprise not only the eye but also the optic nerve and above all the brain.

知道眼睛的用途很简单：粗略地说，眼睛就是照相机。这个比喻很简单，甚至有点简单，但它是相关的，而且相当正确。关于我们的眼睛如何工作、它们在胚胎阶段的发育以及导致它们出现的进化过程，有许多尚未解决的科学问题。这些都是合理而困难的问题。但我们已经知道得足够多了，以至于声称眼睛是一个谜是荒谬的。

Knowing what eyes are used for is simple enough: roughly speaking, they’re cameras. The metaphor is simple, and even a bit simplistic, but it’s relevant and reasonably correct. There are many unresolved scientific questions about how our eyes work, their development in the embryonic stage, and the evolutionary processes that led to their emergence. These are legitimate and difficult questions. But we know enough to have reached the point where it’d be ridiculous to claim that eyes are something of a mystery.

至于视神经，可以将其视为连接眼睛和大脑的一种电缆。在这里，这个比喻再次非常恰当（尽管视神经也执行一些信号预处理）。

As for the optic nerve, it can be seen as a kind of cable linking the eye to the brain. Here again the metaphor works quite well (although the optic nerve does also perform some signal preprocessing).

另一方面，一旦穿过视神经进入我们的视觉皮层，就会出现一个非常有趣的现象，而我们长期以来一直无法理解它。你可以理直气壮地说，它长期以来一直是科学史上最大的谜团之一。

On the other hand, what happens once you get past the optic nerve and inside our visual cortex is a very intriguing phenomenon that has long resisted our efforts to understand it. You can legitimately say that it has long been one of the great mysteries in the history of science.

相机的图像由像素组成：像素是一个网格，每个方格都具有特定的红、绿、蓝亮度值。谜题在于我们的大脑如何处理从视神经接收到的原始信息，以“提取含义”并“识别”图像中的内容。

With a camera, an image is formed of pixels: it’s a grid in which each box has a certain value of red, green, and blue luminosity. The mystery concerns the way our brain treats the raw information received from the optic nerve to “extract meaning” and to “recognize” what is in the image.

总结这个问题的一个好方法是：通过了解所有像素的亮度和颜色值，你怎么知道图像中某处有大象？

Here is a good way to sum up the problem: by knowing the luminosity and the color values of all the pixels, how is it that you know there’s an elephant somewhere in the image?

“大象性”的概念

The Concept of “Elephantness”

我们很容易就识别出大象，但这更令人不安，因为正如我们在上一章中看到的，我们无法真正定义它们是什么。

The ease with which we recognize elephants is all the more troubling since, as we’ve seen in the previous chapter, we’re unable to really define what they are.

这并非巧合。我们想用我们感知到的方法来定义大象，因为这种定义对我们来说最有意义。但我们的大脑用来识别大象的方法非常高效，但完全无法用语言表达出来。

This is no coincidence. We’d like to define elephants as we perceive them, because that definition would have the most sense for us. But the method that our brain uses to recognize elephants is at the same time stunningly efficient and perfectly impossible to translate into words.

事实上，它效率之高令人难以置信。

In fact, it’s so efficient that it’s hardly believable.

首先，你识别大象的能力并不取决于你观察的角度。无论你是从前面还是后面看它，从侧面还是四分之三的视角看它，无论它是直立还是躺着，无论它是大还是小，无论它如何相对于你自己的运动，你都能立即认出它。你对大象可能表现出的众多异常和意想不到的特征也有着令人难以置信的容忍度。无论大象是白化病还是涂有几何形状或彩色条纹，你仍然会知道它是一头大象。

First off, your ability to recognize an elephant doesn’t depend on the angle you’re looking from. Whether you see it from in front or from behind, from the side or in three-quarters view, standing upright or lying down, whether it’s big or small, however it’s moving in relation to your own movement, you recognize it instantly. You also have an incredible tolerance for a multitude of abnormalities and unexpected characteristics an elephant might exhibit. Whether the elephant is albino or painted with geometric shapes or colored stripes, you’ll still know it’s an elephant.

然而，从严格的视觉角度来看，即从原始图像中像素的亮度和颜色的角度来看，这些异常情况确实没有太多共同之处。

From a strictly visual point of view, however, that is, from a point of view of the luminosity and color of the pixels in the raw image, these anomalous situations really don’t have much in common.

但这还不是全部。当一个孩子第一次看到一头真正的大象时，即使他们以前从未见过照片或听说过它，如果你用手指着大象说：“那是一头大象”，孩子也会立即知道你在说什么。

But that’s not all. When a child sees a real elephant for the first time, even if they’ve never seen a picture or heard it spoken of before, if you point your finger at the elephant and say, “That’s an elephant,” the child knows immediately what you’re talking about.

这并不像看上去那么明显。是什么让孩子不去想你所说的大象只不过是它的左前脚、象鼻、象鼻的一部分，或是坐在象鼻上的一只苍蝇呢？

That’s not as obvious as it may seem. What keeps the child from thinking that what you’re calling an elephant is simply the left front foot, or the trunk, or a piece of the trunk, or a fly sitting on the trunk?

如果孩子马上就明白了，那是因为他们已经看到了大象。在你说出那是什么之前，他们就立刻注意到了它。大象是值得命名的非凡之物。他们可能正准备问你那是什么。

If the child understands right away, it’s because they already see the elephant. They noticed it immediately, well before you said what it was. The elephant stood out as something remarkable that deserved a name. They were probably getting ready to ask you what it was.

如果没有这种能力，我们的语言就根本不会存在。我们将无法解释词语所指的内容。

Without this ability, our language simply wouldn’t exist. We wouldn’t be able to explain what words referred to.

还有更令人惊讶的事情。如果你的孩子不是第一次亲眼见到大象，而是从怪诞的卡通画开始，那不会有问题。他们会完全相信他们知道大象是什么。当他们真正看到大象的那一天，他们可能会对大象的大小感到惊讶和害怕，但他们会很容易认出他们已经认识的动物。

There’s something even more surprising. If, instead of meeting elephants for the first time in the flesh, your child began by seeing grotesquely cartoonish drawings of them, that wouldn’t be a problem. They’d be perfectly convinced they knew what an elephant was. The day they saw one for real they might be surprised and probably intimidated by its size, but they’d easily recognize the animal they already knew.

所有这些的结论是，我们的大脑似乎会自动从视神经不断输入的原始视觉数据中提取出大象的普遍概念。然后，它就能够通过大象的多种化身来识别大象这个抽象概念，而大象的化身如此多样，以至于试图将它们全部列出来是荒谬的。

The conclusion to all of this is that our brain seems to automatically extract, from the raw visual data fed continuously into it by the optic nerve, a universal idea of what an elephant is. It then becomes able to recognize this abstract concept of an elephant through its multiple incarnations, in situations so remarkably varied it would be ridiculous to try to list them all.

无需刻意，只需接触到涉及大象的场景，我们就能像变魔术一样，形成一种令人好奇的、可靠的“大象感”。

Without trying, as if by magic, we develop a curiously reliable sense of “elephantness” by mere exposure to scenes involving elephants.

在这个过程的开始，大象只是一个混合了熟悉和奇异的奇怪印象。我们看到它是一头这种动物鼻子大得惊人，腿像树干，耳朵像巨大的扇子。但这种动物并不像我们所知的任何动物。它引起了我们极大的兴趣，我们觉得它应该有一个名字。

At the start of this process, the elephant is nothing but a strange impression mixing the familiar and the bizarre. We see that it’s an animal, with a surprisingly large nose, legs like tree trunks, and ears like giant fans. But this animal doesn’t resemble any we know. It greatly intrigues us and we feel it deserves a name.

大象的概念首先以这种印象的形式出现在我们的视觉皮层中。随着反复观察，图像逐渐稳定并变得更加清晰。在这个过程结束时，大象对我们来说变得如此自然和熟悉，就好像我们一直都知道它们是什么一样。

The concept of elephant first emerges in our visual cortex in the form of this impression. With repeated observations, the image stabilizes and becomes clearer. At the end of the process, elephants become so natural and familiar to us that it’s as if we’ve always known what they are.

但是概念是什么？我们为什么要用概念来思考？它们存在于现实的哪个层面？它们是由什么构成的？是什么让我们能够感知它们？

But what is a concept? Why do we think with concepts? At what level of reality do they exist? What stuff are they made of? What makes us able to perceive them?

这些问题属于哲学史上最古老的问题之一，几千年来，人们一直认为这些问题无法解决。问这些问题当然就是问我们的大脑是如何运作的。

These questions are among the most ancient in the history of philosophy and, for millennia, they have been said to be insoluble. Asking these questions, of course, is asking how our brains work.

有趣的是，视觉之谜重新定义了这些问题，无需抽象或令人困惑的语言。这是一种非常实用、非常务实的方式来探究我们智力的内部运作。

Interestingly, the mystery of vision reframes these questions without any need for abstract or confusing language. It’s a very practical, very down-to-earth way of asking questions about the inner workings of our intelligence.

最糟糕的比喻

The Worst Metaphor

我们经常把大脑比作计算机。这个比喻在两个方面是正确的：大脑和计算机都能够完成复杂的信息处理任务，并且它们都利用电信号。

Our brain is often compared to a computer. This metaphor is correct in two aspects: both the brain and computers are capable of accomplishing complex tasks of information processing, and they both make use of electrical signals.

至于其他的一切，这个比喻是极其错误的。它毁掉了我们理解正在发生的事情的机会。

As for all the rest, the metaphor is catastrophically false. It ruins our chance of understanding what is going on.

计算机是系统 2 的完美体现：它是一种能够以惊人的速度机械地应用长序列逻辑指令而不会出错的机器——这是我们的大脑完全无法做到的。

A computer is a perfect embodiment of System 2: it’s a machine capable of mechanically applying long sequences of logical instructions at breathtaking speeds without making mistakes—something our brain is entirely unable to accomplish.

计算机由中央处理器和存储单元组成，中央处理器负责计算，存储单元负责存储信息。在这些不同的单元之间，信息沿着电路高速流通，无需转换。在我们的大脑中，情况恰恰相反。信息流通缓慢，并在流通的每一步中发生转换。记忆、处理和流通是密不可分的。

A computer is made of a central processing unit where calculations are made and memory units where information is stored. Between these distinct units, information circulates at high speeds along electrical circuits without being transformed. In our brains, it’s quite the contrary. Information circulates slowly and is transformed along each step of its circulation. Memory, processing, and circulation are indissoluble.

计算机按顺序逐条执行指令，由每秒滴答数十亿次的内部时钟控制。激活神经元之间的连接所需的时间约为千分之一秒。因此，我们大脑的基本操作比计算机慢一百万倍。但我们的大脑不是按顺序执行的：它并行处理数十亿个这样的操作。

A computer strings instructions one after the other, sequentially, paced by an internal clock that ticks billions of times per second. The time it takes to activate a connection between neurons is on the order of a thousandth of a second. The base operations of our brain are thus a million times slower than that of a computer. But our brain isn’t sequential: it processes in parallel billions and billions of these operations.

计算机的硅电路是不可改变的，刻在惰性材料上。我们的大脑是活组织，会不断自我重构。

The silicon circuits of computers are immutable, engraved in an inert material. Our brain is living tissue that constantly reconfigures itself.

感知系统

A Perceptual System

与其将大脑视为一个计算系统，不如将其视为一个感知系统，这样更有启发性。我们的大脑是我们感知世界的中央器官。它让我们能够感知事物：例如，我们面前有一头大象。

Rather than representing the brain as a system for doing calculations, it’s much more illuminating to see it as a perceptual system. Our brain is the central organ through which we perceive the world. It allows us to sense things: for example, that an elephant is in front of us.

我们大脑中的每个神经元本身都是一个微小的感知系统。从解剖学角度来看，典型的神经元由三部分组成：

Each neuron in our brain is in itself a tiny perceptual system. Anatomically, a typical neuron consists of three parts:

— 一种树状结构，分支出数千个称为树突的小受体。树突是神经元的接收端。

—A treelike structure that branches out in thousands of small receptors called dendrites. The dendrites are the receiving end of the neuron.

— 中心部分称为胞体：这是神经元的主体，包含细胞核。

—A central part called the soma: this is the body of the neuron and contains the nucleus.

— 一种称为轴突的树干或茎，它分支出来，末端称为轴突末端。这是神经元的传递端，负责与其他神经元进行通信。

—A kind of trunk or stem called the axon that branches out and ends in what are called axon terminals. This is the transmitter end of the neuron, which communicates with other neurons.

神经元以非常特殊的方向相互连接：一个神经元的树突插入其他神经元的轴突末端，使其能够从中收集信息。这样形成的连接称为突触。

Neurons are connected to one another in a very specific orientation: the dendrites of one neuron are plugged into the axon terminals of other neurons, enabling it to collect information from them. The connections thus formed are called synapses.

神经元表现出一种全有或全无的行为：它们要么处于静止状态，要么突然全力“启动”，在这种情况下，电动作电位会沿着它们的轴突传递到终端，触发称为神经递质的分子释放到突触中。

Neurons exhibit an all-or-nothing type of behavior: they can either be in a resting state or suddenly “fire up” in full force, in which case an electrical action potential travels down their axon to the terminals, triggering the release into the synapses of molecules called neurotransmitters.

这些神经递质反过来又被接收神经元的树突检测。

These neurotransmitters are in turn detected by the dendrites of receiving neurons.

为了决定是否应该启动，神经元本质上会进行一次轮询。如果足够多的树突检测到上游神经元刚刚启动，神经元本身就会启动，这反过来又可能触发下游神经元的启动。

To decide whether it should fire up, a neuron essentially conducts a poll. If enough of its dendrites detect that upstream neurons have just fired up, the neuron will itself fire up, which in turn may trigger the firing up of downstream neurons.

神经元是一个具有狭隘世界观和二元反应的感知系统：它从世界中感知到的一切都是大脑的活动。位于上游的神经元，除了休息，它所能做的就是启动。

A neuron is a perceptual system with a narrow worldview and a binary response: all it perceives from the world is the activity of the neurons that are immediately upstream and, apart from resting, all it can do is fire up.

一个新兴的属性

An Emergent Property

第一次有人试图向我解释神经元是如何工作的时，我一点兴趣都没有。似乎没有任何线索。如果我们的神经元如此原始，我们怎么会变得聪明呢？

The first time someone tried to explain to me how neurons worked, it didn’t interest me at all. It didn’t seem to lead anywhere. If our neurons are so primitive, how can we be intelligent?

只要你试图将我们的智力机制定位在我们大脑的某个特定位置，你就不可能理解它们。智力是一种所谓的涌现属性：我们的神经元是原始的和有限的，但大量的神经元集合使令人难以置信的复杂行为“涌现”，这些行为本身无法归因于任何一个神经元——这些大规模行为就是我们所说的智力。

The mechanisms of our intelligence are impossible to understand as long as you try to locate them in a specific place in our brain. Intelligence is what is called an emergent property: individually our neurons are primitive and limited, but vast assemblies of neurons make incredibly sophisticated behaviors “emerge” that can’t be attributed to any neuron by itself—these large-scale behaviors are what we call intelligence.

这有点像交通堵塞：你可以花二十年时间对汽车进行逆向工程，但这并不能让你学到任何有关交通堵塞的知识。然而交通堵塞确实存在，而且完全是由汽车造成的。

It’s a bit like traffic jams: you can spend twenty years of your life reverse-engineering cars, but that won’t teach you anything about traffic jams. And yet traffic jams exist and they’re entirely made up of cars.

我们神经元的个体行为和大脑的整体功能之间存在着巨大的鸿沟。长期以来，这种鸿沟似乎如此巨大，以至于科学家们对理解它感到绝望。

An enormous divide separates the individual behavior of our neurons and the overall functioning of our brain. For a long time, this divide seemed so enormous that scientists despaired of ever understanding it.

但现在情况已不再如此。视觉之谜现在已基本得到解决。我们无法理解一切，但我们所理解的内容足够详细，也足够有意义，以至于对某些人来说，它不再感觉像是一个谜（尽管当然，许多深层次的问题仍未得到解决）。

That’s no longer the case. The mystery of vision is now largely resolved. We don’t understand everything, but what we understand is sufficiently detailed and makes enough sense that, to some people, it no longer feels like there is a mystery (although, of course, many deep questions remain open).

首先，得益于神经学的进步，我们现在对大脑的整体组织和神经元的接线图有了更好的了解。现在我们也可以跟踪实时监测人类和动物大脑中单个神经元或特定区域的活动。

First, thanks to progress in neurology, we now have a much better understanding of the overall organization of our brain and the wiring diagram of our neurons. It has also become possible to follow in real time the activity of individual neurons or specific regions of the brain in both humans and animals.

这本身并不能解决这个谜团。脑成像技术仍然无法以足够的分辨率描绘出整个大脑活动，从而无法完全了解正在发生的事情。我们距离同时跟踪所有神经元在识别大象时的工作情况还很远，甚至距离能够在整个终身学习过程中跟踪这些神经元的情况更远。

This, in itself, wouldn’t solve the mystery. Brain-imaging technologies are still incapable of mapping out the entire brain activity with enough resolution to fully understand what’s going on. We’re still very far from being able to simultaneously follow, for example, all the neurons at work in recognizing an elephant—and even further from being able to follow these neurons throughout the lifelong process of learning.

最引人注目、最引人注目的突破来自另一个学科：计算机科学。自 20 世纪 50 年代以来，心理学家和计算机科学家一直在从我们的神经元功能和大脑皮层解剖结构中寻找灵感，以构建人工智能系统。由于这些系统模仿了我们大脑的结构，因此它们的行为可以揭示我们大脑中正在发生的事情。

The most spectacular and compelling breakthrough has come from another discipline: computer science. Since the 1950s psychologists and computer scientists have looked for inspiration in the functioning of our neurons and the anatomy of our cerebral cortex to construct systems of artificial intelligence. Because they imitate the architecture of our brain, the behavior of these systems sheds light on what’s going on in our heads.

弗兰克·罗森布拉特 (Frank Rosenblatt，1928-1971) 是这种方法的先驱之一，他帮助构建了第一个神经元数学模型，并设计了实现该模型的计算设备。但对能够模拟我们视觉能力的复杂神经网络的行为进行建模是一个完全不同规模的问题。这项技术跌跌撞撞地发展了几十年，经历了无数的起起落落。在某个时候，人工智能社区变得如此失望，以至于人工神经网络被视为技术死胡同。三位科学家 Geoffrey Hinton、Yann LeCun 和 Yoshua Bengio 继续相信这种方法。历史证明他们是正确的。

Frank Rosenblatt (1928–1971), one of the pioneers of this approach, helped construct the first mathematical model of a neuron and fashioned a computing device that implemented this model. But modeling the behavior of complex neural networks capable of simulating our ability to see was a problem of an entirely different scale. The technology stumbled along for decades and went through numerous ups and downs. At some point, the AI community grew so disillusioned that artificial neural networks were seen as a technological dead end. Three scientists, Geoffrey Hinton, Yann LeCun, and Yoshua Bengio, continued to believe in the approach. History proved them right.

到 21 世纪末，他们的“深度学习”算法取得了巨大进步，能够解决图像识别的高级问题，例如检测大象的存在。

Toward the end of the 2000s their “deep-learning” algorithms had made so much progress that they had become capable of resolving advanced problems in the recognition of images, such as the detection of the presence of elephants.

有效的比喻

An Effective Metaphor

2010 年左右，当我开始熟悉这些算法时，我很高兴第一次发现了一种描述理解过程的方式，这种方式与我亲身经历的相兼容。

Around 2010, when I began to familiarize myself with these algorithms, I was excited to discover a way of describing the process of understanding that, for the first time, was compatible with what I had personally experienced.

从本书一开始，我就讨论了一些强大而神秘的现象，这些现象在我的数学生涯中一直困扰着我：思维的可塑性，人类语言不可避免的模糊性，时间和反复试验在试图理解事物中的作用，提出愚蠢问题的必要性，以及事后显而易见的感觉。

From the outset in this book, I’ve discussed a number of powerful and mysterious phenomena that have perplexed me throughout my mathematical career: mental plasticity, the inevitable ambiguity of human language, the role of time and trial and error in trying to understand things, the necessity to ask stupid questions, the feeling of obviousness that comes after the fact.

有了深度学习，理解的过程就可以变得有形而具体。我们终于可以在不借助某种黑魔法的情况下谈论它了。

With deep learning, the process of understanding could be made tangible and concrete. It had finally become possible to speak of it without invoking some kind of black magic.

我对这个主题如此着迷，以至于我决定结束我的数学生涯。我刚刚完成了代数和几何研究的一个重要周期，我看到了一个探索全新主题的机会，这个主题也许能够阐明我的经历。

The subject fascinated me so much that I decided to call an end to my career in mathematics. I had just completed an important cycle in my research in algebra and geometry, and I saw an opportunity to explore a radically new theme, one that might be able to shed light on what I’d experienced.

我选择以最实际的方式来解决这个问题：辞去学术职位，创办一家人工智能初创公司。

I chose to approach it in the most practical manner possible, by quitting my academic position and founding an artificial intelligence startup.

为了解释我们的智力的本质和我们的思维机制，深度学习提供了我所知道的最好的比喻。

To explain the nature of our intelligence and the mechanisms of our thought, deep learning offers the best metaphor I know of.

大象的神经元

An Elephant Neuron

深度学习让你解开的第一个谜团是概念的出现。换句话说，千百年来形而上学中最激烈的争论之一突然在软件领域重现，让我们面对一个无可争辩的实验现实：概念思维自发出现在大量人工神经元中，存在非结构化数据，例如大量图像。

The first mystery that deep learning allows you to dissipate is that of the emergence of concepts. In other words, what had been for millennia one of the liveliest debates in metaphysics was suddenly reincarnated in the realm of software, confronting us with an undisputable experimental reality: conceptual thought spontaneously emerges in vast assemblies of artificial neurons subjected to unstructured data, for example, a flood of images.

粗略地说，在视觉环境下，它的工作原理如下。深度学习算法将我们的大脑皮层建模为一个具有多层的神经网络。第一层是原始图像：一个代表像素的神经元矩阵。第二层由树突与第一层神经元相连的神经元组成。第三层由树突与第二层神经元相连的神经元组成，依此类推。正是因为网络由许多叠加的层组成，所以它被称为“深度”学习。

Roughly speaking, here’s how it works in the context of vision. Deep-learning algorithms model our cortex as a neural network with multiple layers. The first layer is the raw image: a matrix of neurons that represent pixels. The second layer is formed of neurons whose dendrites are linked with neurons in the first layer. The third layer is formed of neurons whose dendrites are linked to the neurons in the second layer, and so on. It’s because the network is made up of many superimposed layers that it’s called “deep” learning.

在描述神经元如何运作时，我忽略了一个重要细节：当神经元对其树突进行投票以决定是否应该启动时，投票并不民主。神经网络中的每个连接都带有一定的“权重”，这决定了它对决策的影响有多大。

In my description of how neurons function, I omitted one important detail: when a neuron runs a poll of its dendrites to decide whether it should fire up, the poll isn’t democratic. Each connection in a neural network carries a certain “weight” that determines how much it counts toward the decision.

当网络受到大量原始图像的影响时，它会根据我将在后面几页解释的机制逐渐调整所有权重。

When the network is subject to a flood of raw images, it gradually adjusts all the weights according to a mechanism I’ll explain in a few pages.

正是通过这个调整权重的过程，网络才能“学习”并“变得智能”。

It’s through this process of adjusting weights that the network “learns” and “becomes intelligent.”

例如，当你让深度学习算法长时间运行，让它从互联网上随机拍摄的数以百万计的照片中“学习”，你会注意到每个神经元逐渐变得专注于检测某个“概念”。

When you let a deep-learning algorithm run for a long time, for example, by making it “learn” from millions and millions of photos taken at random from the internet, you notice that each neuron gradually comes to specialize in the detection of a certain “concept.”

前几层的概念非常原始，而更深层的概念则要复杂得多。

The concepts of the first layers are very primitive, while those of the deeper layers are much more sophisticated.

例如，第二层中的神经元可能专门检测图像左下角的垂直线，或图像另一区域中亮度的细微变化。当此元素存在时，它将专门激活。

For example, a neuron in the second layer might specialize in the detection of a vertical line in the bottom left corner of an image, or in the slight change in luminosity in another region of the image. It will fire up exclusively when this element is present.

在第三层，概念变得稍微复杂一些例如，一个神经元可能检测出位于图像特定区域的两个部分之间的特定类型的角度。

In the third layer the concepts become slightly more sophisticated. For example, a neuron might detect certain types of angles between two segments situated in a certain zone of the image.

随着你进一步深入网络，概念变得越来越丰富和抽象。它们变得越来越“深”。在第五层，某些神经元可能专门用于检测三角形或某些类型的曲线。

As you get further into the network, the concepts grow in richness and abstraction. They become more and more “deep.” In the fifth layer, certain neurons might, for example, specialize in the detection of triangles or certain types of curves.

在第二十层，神经元可能专门用于检测大象——无论它们是真实的还是画出来的。

In the twentieth layer, a neuron might specialize in the detection of elephants—whether they’re real or drawn.

这种呈现方式是故意简化的。现实情况比这更复杂，神经元和概念之间的对应关系不一定那么直接。具体实验确实表明，对于你认识的每位著名演员，你确实有一个特定的神经元，它会对他或她在屏幕上的出现做出特定反应（参见“注释和进一步阅读”部分）。但一些科学家认为概念对应于神经元组而不是单个神经元。在计算机模拟中，单个神经元有时专门用于检测高级物体，例如大象，但在其他情况下，这种检测会调动整个神经元组，而不是特定的神经元。

This way of presenting things is purposely simplistic. The reality is more complicated than that, and the correspondence between neurons and concepts isn’t necessarily that direct. Specific experiments do suggest that, for each famous actor or actress you know, you really have a specific neuron that reacts specifically to his or her presence on the screen (see “Notes and Further Reading” section). But some scientists think that concepts correspond to groups of neurons rather than individual neurons. In computer simulations, individual neurons sometimes specialize in the detection of high-level objects such as elephants, but in other instances this detection mobilizes a whole group of neurons rather than a specific one.

虽然这个模型过于简单，但它确实捕捉到了我们认知过程的一些最突出的方面。说你的大脑中有一个专门的大象神经元有点夸张，但它很有启发性。这就是为什么我们会假设这是真的。

While simplistic, this model does capture some of the most salient aspects of our cognitive processes. Saying that in your brain there is a dedicated elephant neuron is a bit of a stretch, but it’s an illuminating one. This is why we’ll proceed as if it really were the case.

一百万亿根细丝

One Hundred Trillion Filaments

你的大象神经元有成千上万个树突。你对大象的个人定义涉及成千上万个标准，这些标准本身就是相当高层次的抽象属性，例如“是动物”、“有鼻子”、“有大耳朵”、“是灰色的”、“很大”、“有喇叭”、“有象牙”、“皮肤粗糙”、“以这样或那样的方式移动”等等。

Your elephant neuron has thousands of dendrites. Your personal definition of an elephant involves thousands of criteria, which are themselves abstract attributes at a fairly high level, such as “is an animal,” “has a trunk,” “has big ears,” “is gray,” “is big,” “trumpets,” “has ivory tusks,” “has rough skin,” “moves in such and such manner,” and so on.

这些属性中的每一个都有自己的权重。“有象鼻”的权重很高，这是可以肯定的，因为有象鼻是一个高度相关的特征。你的大象神经元通过把神经元被激活的属性的权重相加来计算出“大象性得分”。

Each of these attributes has its own weight. It’s a good bet that “has a trunk” has a high weight, because having a trunk is a highly relevant feature. Your elephant neuron calculates an “elephantness score” by adding up the weights of the attributes whose neurons are activated.

一旦分数超过某个阈值，神经元就会判定你面对的是一头大象。低于这个阈值的地方是灰色区域，你不确定它是不是大象（其他人可能有不同的看法），然后是灰色区域，你肯定不是大象。

Once the score passes a certain threshold, the neuron decides that you’re dealing with an elephant. Beneath this threshold there’s a gray zone where you’re not sure it’s an elephant (and where someone else might have a different opinion), then a zone where it’s clearly not an elephant.

正是大量的标准使得你的大象检测系统如此强大和可靠。你的大象分数经过充分采样，在不可预见的情况下仍然适用，并能容忍各种各样的异常情况。

It’s the large number of criteria at play that makes your elephant-detection system so robust and reliable. Your elephantness score is sufficiently well sampled to stay relevant in unforeseen situations and to tolerate a large variety of anomalies.

显然，准确的定义是不可能写出来的。写一整本书也不够，而且无论如何你也找不到合适的词句。

The exact definition is evidently impossible to write. An entire book wouldn’t be enough, and at any rate you’d never find the words.

这 100 万亿个神经连接纠缠在一起，就是维特根斯坦所说的蜘蛛网。解开这一切是不可想象的。但如果不解开这一切，就不可能定义任何东西。

This tangle of 100 trillion neural connections is the spider’s web Wittgenstein was talking about. It’s unthinkable to unravel all of that. But without unraveling all of that, there’s no chance of being able to define anything.

深度学习算法，即使是最强大、最复杂的算法，也只是对我们大脑结构的粗略简化。我们的大脑皮层确实是分层结构的，尽管不像计算机模型那样严格和狭窄：你的“大象”神经元会轮询你的“象鼻”神经元，但你的“象鼻”神经元本身肯定会轮询你的“大象”神经元。定义的循环是无法避免的。

Deep-learning algorithms, even the most powerful and sophisticated, are only gross simplifications of our cerebral architecture. Our cortex is indeed structured in layers, though not as strictly and narrowly as in computer models: your “elephant” neuron polls your “trunk” neuron, but your “trunk” neuron itself certainly polls your “elephant” neuron. The circularity of definitions is impossible to avoid.

认为我们的大脑由彼此完全隔离的专门区域组成也过于简单了：视觉发生在更广泛的范围内，而且无论如何，你对大象的定义并不完全是视觉的。

It is also an oversimplification to imagine that our brains are organized in specialized regions perfectly isolated from each other: vision happens within a broader context and, at any rate, your definition of elephant isn’t entirely visual.

有机的学习过程

The Organic Process of Learning

剩下的就是描述学习过程本身：神经元依靠什么机制来确定它们与其他神经元的连接的“权重”？

It remains to describe the process of learning itself: what mechanisms do the neurons rely on to determine the “weight” of their connections with other neurons?

让我们回到大象神经元的例子。它不断分析上游神经元的状态，以决定是否应该启动。你正在实时审视世界，警惕大象。（说“实时”总是滥用语言，因为没有一个系统真正实时运行。一个神经元需要大约半毫秒才能启动。）

Let’s go back to the example of your elephant neuron. It is constantly analyzing the state of its upstream neurons to decide whether or not it should fire up. You’re scrutinizing the world in real time, on the watch for elephants. (It’s always an abuse of language to speak of “real time,” because no system actually functions in real time. It takes a neuron around half a millisecond to fire up.)

在第 11 章中，我们将其称为系统 1，即瞬时直觉思维，它让您感觉思考速度如闪电般快。

In chapter 11, we called this System 1, instantaneous intuitive thinking, that which gives you the impression of thinking as fast as lightning.

与此同时，另一种现象也在幕后发生。它发生的速度要慢得多，而且非常离散，我们无法察觉。正确的比喻不是闪电，而是有机生长。这是我们学习的过程。这是我们所说的系统 3 的基础，即我们逐渐改变我们向自己呈现世界的方式的能力。

In parallel to this, another phenomenon takes place in the background. It happens at a much slower pace, and is so discrete that we can’t perceive it. The correct metaphor isn’t lightning, but organic growth. It’s the process through which we learn. It’s the basis of what we have called System 3, our ability to gradually modify the way we represent the world to ourselves.

如果有一天你遇到一头没有鼻子的大象，你会感到惊讶。

If one day you come across an elephant without a trunk, you’d be surprised.

“惊讶”是什么意思？一头无鼻子的大象让你惊讶，因为你对世界的认识没有预料到它。但这仍然不会阻止你理解。你几乎肯定会看到它是一头大象，同时又有一种不安的感觉，觉得有什么地方不对劲。

What does it mean “to be surprised”? A trunkless elephant surprises you because your vision of the world hadn’t anticipated it. That still doesn’t keep you from understanding. You’d almost certainly still see that it’s an elephant, while having the disturbing feeling that something’s terribly wrong.

当你用数学方法对深度学习系统进行建模时，你可以定义一个数值来衡量其在特定情况下的“困惑度”。学习系统会调整其权重以降低其困惑度。

When you mathematically model a deep-learning system, you can define a numerical quantity that measures its “perplexity” in a given situation. A system that learns is one that adjusts its weights in order to reduce its perplexity.

直观地看，这就是困惑的意思。在你的大象性评分中，“有象鼻”的权重相当高，因为大象和象鼻通常一起出现。虽然其他标准可以让你在没有象鼻的情况下进行补偿并“看到”大象，但这是一种不正常的情况，无论你是否意识到，你都会有身体上的感受。

Intuitively, this is what perplexity means. In your elephantness score, “having a trunk” carries a pretty high weight, because elephants and trunks usually come together. While the other criteria will allow you to compensate and “see” an elephant despite the absence of a trunk, it’s an abnormal situation and you feel it physically, whether you’re aware of it or not.

你的大象神经元感到困惑。它检测到了大象，但没有象鼻，而在此之前，象鼻被认为是一个基本特征。神经元通过稍微减少相关权重来考虑这一新现实。如果你继续遇到没有象鼻的大象，你最终几乎不会考虑这个标准。

Your elephant neuron is perplexed. It detected an elephant, yet there was no trunk, and up to this point trunks were supposed to be an essential feature. The neuron factors in this new reality by slightly diminishing the associated weight. If you continue to come across elephants without trunks, you’d end up by hardly taking this criterion into account at all.

事实上，神经元没有必要出现如此大的异常来纠正它们的权重。它们会随着每次刺激而不断调整，即使调整幅度很小。从生理学上讲，这对应于突触连接增强或减弱的能力。新的连接会建立，其他连接会消失。我们的心理回路会不断自我重新配置。

In reality there’s no need to have such large abnormalities for your neurons to correct their weights. They’re constantly, if slightly, in adjustment with each stimulation. Physiologically, that corresponds to the ability of synaptic connections to strengthen or weaken. New connections are created and others disappear. Our mental circuitry is constantly reconfiguring itself.

心理可塑性不过如此：你的神经元的分散活动，每个神经元都试图加强其分数的一致性。

Mental plasticity is nothing more than this: the decentralized action of your neurons that, individually, seek to reinforce the consistency of their score.

最不寻常的事情是——这已通过深度学习算法在实验中得到完美证明——这种简单的机制允许高级抽象概念（如大象）从随机选择连接和权重的出发状态开始逐渐出现。

The most extraordinary thing—and it’s been perfectly demonstrated experimentally thanks to deep-learning algorithms—is that such simple mechanisms allow high-level abstract concepts, such as elephants, to gradually emerge, starting from a state of departure where the connections and weights are chosen at random.

你并非生来就拥有大象神经元。第一次看到大象神经元时，你大惑不解：你的“动物”神经元很兴奋，你的“值得我全神贯注的巨大事物”神经元也很兴奋，还有许多其他神经元，它们与你能识别的许多属性相对应。但这种强大而复杂的神经元压力没有名字。你仔细观察它，吸收它，学习它。

You weren’t born with an elephant neuron. The first time you saw one, you were greatly perplexed: your “animal” neuron was excited, as well as your “enormous thing that deserves my full attention” neuron, as well as many other neurons corresponding to the many attributes you could recognize. But this powerful and complex impression had no name. You looked at it carefully, to take it in and to learn.

在你的脑海里，大象最初是一个复合体，调动了你大脑中 1000 亿个神经元中的很大一部分。在第一次遇到大象时激活的其中一个神经元有着特殊的命运。一点一点地，通过逐渐调整其权重，它变得越来越专业化。随着时间的推移，它成为了你的大象神经元。

In your head, the elephant was at first a composite object, mobilizing a large number among the 100 billion neurons in your brain. One of the neurons that fired up during this first elephantine encounter had a special destiny. Little by little, by gradually adjusting its weights, it became more and more specialized. Over time, it became your elephant neuron.

深度学习网络中的概念是在接触世界后产生的。它们实际上是凭空而来的，就像风在平坦的海洋上形成波浪一样：最初，水面上只有微小的随机不规则性，但这些不规则性随后被反馈机制放大，遵循物理定律，这些定律在微观层面上很容易描述，而在更大的尺度上，则会产生极其复杂的突发现象。

Concepts emerge in deep-learning networks under the simple effect of exposure to the world. They emerge literally out of nothing, as waves are formed on a flat ocean by the effect of the wind: initially, there are just tiny random irregularities on the surface of the water, but these irregularities are then amplified by feedback mechanisms, following laws of physics that are simple to describe on the microscopic level and that, on a greater scale, give rise to incredibly complex emergent phenomena.

抽象而模糊

Abstract and Vague

所有这些的科学、技术和哲学含义远远超出了本书的范围。以下是我们根据我们的需求总结的。

The scientific, technological, and philosophical implications of all this go far beyond the scope of this book. Here’s how we can sum it up as far as our needs go.

我们的大脑和任何动物的大脑一样，都是一个感知机器，不断制造抽象概念。我们通过错综复杂的神经连接网络构建并维持物质世界的表征。这种对世界的表征是一层又一层抽象概念的堆积。从本质上讲，它本质上是概念性的。

Our brain, like any animal brain, is a perceptual machine that constantly fabricates abstractions. We construct and we maintain a representation of the material world through the tangled network of our neural connections. This representation of the world is a piling up of layers upon layers of abstractions. Down to its very core, it’s conceptual in nature.

概念思维不是人类的特权。它不是源自我们的语言或文化。当我这样说时，我使用“思维”这个词的含义非常广泛，指的是神经过程构成我们智力的基础。任何狮子都以概念的方式思考，并且其头部有一个大象神经元。

Conceptual thought isn’t a human privilege. It doesn’t arise from our language or our culture. When saying this, I’m using the word thought in a very broad sense, to designate the neurological processes that constitute the substrate of our intelligence. Any lion thinks in a conceptual manner, and has an elephant neuron in its head.

我们语言的缺陷只是其神经基础的反映。我们赋予词语的意义是感知性的：我们知道如何识别大象，但我们永远无法真正定义它是什么。

The flaws of our language are but a reflection of its neurological underpinnings. The meanings that we assign to words are perceptual: we know how to recognize an elephant but we can never really define what it is.

任何定义都是近似的。词语的含义总是流动的、模糊的、变化的。没有什么是明确的。在我们的头脑里，世界是抽象而模糊的。

Every definition is an approximation. The meaning of words is always fluid, ambiguous, changing. Nothing is ever clear-cut. Inside our head, the world is abstract and vague.

20

数学的觉醒

20

A Mathematical Awakening

在我的整个数学旅程中，尽管我热爱数学并享受数学给我带来的乐趣，但我总是觉得真正的挑战在别处。

All throughout my mathematical journey, despite my taste for math and the pleasure it brought me, I’ve always had the impression that the real challenge was elsewhere.

真正重要的、激励我并让我想要继续下去的，不是我能证明的定理，也不是只有少数专家感兴趣的定理，而是其他东西。这其他东西更加深刻，更加普遍。

What really mattered, what motivated me and made me want to continue, wasn’t the theorems that I could prove and that would interest only a few specialists, but something else. This other thing was much more profound and much more universal.

它甚至显得非常重要。但我却无法真正解释它是什么，甚至无法给它起名字。

It even seemed to be incredibly important. Yet I couldn’t really explain what it was, and I wasn’t even able to give it a name.

这个问题困扰了我很多年。我有一种奇怪的感觉，许多富有创造力的数学家都熟悉这种感觉，觉得有些事情正在发生，有些事情不明确，需要解释。我不知道那是什么，但我知道它与人类的理解有关。这让我大致知道该往哪个方向走。

This troubled me for many years. I had the weird feeling, which is familiar to many creative mathematicians, that something was going on, something unclear that deserved an explanation. I had no idea what it was, but I knew that it had to do with human understanding. That gave me a rough idea of which direction to follow.

在我看来，数学研究是解决这个问题的最佳途径。我就像一个探险家，出发去探索地图上几乎找不到的未知大陆。我不知道自己会发现什么，但从一开始就很清楚，实际上，我是在寻求发现自己。

Mathematical research seemed to me to be the best way to approach it. I was like an explorer who set out to discover an unknown continent that is barely sketched out on the map. I had no idea what I’d find, but it was perfectly clear from the start that, in reality, I was seeking to discover myself.

这本书讲述了我的冒险经历。我亲身经历了这一切，所以我可以讲述这个故事。

This book is the story of my adventure. I lived through it so I could tell the tale.

很多年过去了，我终于能够用语言表达这件事了当时，这对我来说似乎很奇怪，也不清楚。这就是最后一章的主题。

Years have passed and I’m finally able to put into words the thing that, at the time, seemed so strange and unclear to me. It’s the subject of this final chapter.

从天而降

Fallen from the Sky

当我还是一名博士生时，有人问我我的研究是否有用，我用一个笑话回避了这个问题：“一千年后它将在物理学中得到应用。”

When I was a PhD student and someone asked me about the usefulness of my research, I sidestepped the question with a joke: “It will be used in physics in a thousand years.”

我曾经对当代数学研究的实际应用持怀疑态度。过去二十年改变了我的想法。

I was very skeptical about the practical applications of contemporary mathematical research. The past twenty years have changed my mind.

你日常生活中使用的所有科技产品都是使用高级数学设计和制造的。每一条被记录或远距离传输的信息都得益于复杂的数学处理。每次你与智能手机互动时，你都会与交织在一起的数学抽象堆栈互动。

All the technological objects that you use in your day-to-day life are designed and built using advanced mathematics. Every piece of information that is recorded or transmitted over a distance can only be so thanks to sophisticated mathematical processing. Every time you interact with your smartphone, you interact with interwoven stacks of mathematical abstractions.

几个世纪以来，数学在科学和技术中一直扮演着重要的角色。我们的世界和生活的数字化使这一现象放大了几个数量级。毫无疑问，数学在技术上很有用，而且这种用处日益增加。

For centuries, math has played a prominent role in science and technology. The digitization of our world and our lives has amplified this phenomenon by orders of magnitudes. There’s no possible doubt that math is technologically useful, and getting more and more so by the day.

事实上，数学已经非常有用。

Math already is, in fact, frighteningly useful.

然而，从更广阔的历史视角来看，世界的数学化似乎是一种新现象。它可以追溯到笛卡尔和在他之前伽利略，后者曾宣称宇宙是一本“用数学语言写成的书”。

However, when put in a broader historical perspective, the mathematization of the world appears as a recent phenomenon. It dates back to Descartes and just before him Galileo, who famously declared that the universe was a book “written in the language of mathematics.”

在十七世纪之前，科学并没有数学化。数学实际上没有任何应用。它仍然由算术和几何中的“幼稚和无意义”练习组成，笛卡尔对此很是苦恼。但这并没有阻止古希腊人将其作为哲学的先决条件。

Prior to the seventeenth century, science wasn’t mathematized. Mathematics didn’t really have any application. It still consisted of the “childish and pointless” exercises in arithmetic and geometry that Descartes was so upset about. That hadn’t prevented the ancient Greeks from making it a prerequisite for philosophy.

无意冒犯伽利略，但数学是宇宙语言这一概念对我来说毫无意义。当我听说数学主要通过其在科学和技术中的应用而有用时，我同样持怀疑态度。这种谈论数学的方式让整个科学史完全无法理解：数学家是如何找到一种方法来熟悉宇宙语言的？数学是从天上掉下来的吗？是上帝派来的吗？为什么古希腊人还没有弄清楚它是宇宙的语言，却仍然坚持教授数学？

No offense to Galileo, but the notion that mathematics is the language of the universe doesn’t make any sense to me. I’m equally skeptical when I hear that math is useful primarily through its applications in science and technology. This way of talking about math makes the whole history of science totally incomprehensible: How did mathematicians find a way to get acquainted with the language of the universe? Did mathematics fall from the sky? Was it sent by God? Why did the ancient Greeks, who hadn’t figured out that it was the language of the universe, still insisted on teaching mathematics?

它没有任何实际用途，但是什么促使它在过去几千年里不断发展呢？

What could have motivated its development over those millennia when it served no practical purpose?

真正的数学

The True Mathematics

将数学视为一种外部工具是最能让我们讨厌它的方式。官方数学有着尖锐的棱角、冷酷的逻辑和令人难以忍受的优越感，不可能让人爱上它。但正如我们所见，还有另一种方式。

Presenting mathematics as an external tool is the surest way to make us hate it. Official math, with its sharp edges, its cold logic, its unbearable air of superiority, is impossible to fall in love with. But as we’ve seen, there is another way.

我在这本书中谈到了我如何利用直觉在数学上取得进步。至少一开始我就是这么认为的，当时我还认为官方的数学知识和书本上的内容才是最重要的。

I’ve spoken in this book about how I used my intuition to get ahead in math. At least that’s what I believed I was doing at first, when I still thought that what counted was the official math, the stuff in the books.

随着我逐渐成熟，我意识到事情恰恰相反。我利用数学来培养我的直觉。

As I matured, I came to the realization that it worked the other way around. I was using math to develop my intuition.

数学首先是一种内在工具。它的主要目的是增强人类的认知能力。通过正确的想象力练习，我们能够对数学概念产生直观和熟悉的理解。我们可以利用它们，使它们成为我们身体的延伸。

Math is first and foremost an inner tool. Its main purpose is to enhance human cognition. With the correct exercises of imagination, we have the ability to develop an intuitive and familiar understanding of mathematical notions. We can appropriate them and make them an extension of our bodies.

真正的数学是秘密的数学，它扩展了我们对周围世界的直觉理解。

The true math is the secret math, the one that extends our intuitive understanding of the world that surrounds us.

你已经掌握了这种内在数学。你知道如何在头脑中操纵一个圆圈。你感觉到数字 999,999,999 就在你面前。当你观察世界时，你会识别数字和几何形状。

You already have access to this inner math. You know how to manipulate a circle in your head. You sense the presence of the number 999,999,999 right there in front of you. When you look at the world, you recognize numbers and geometric shapes.

数学概念在大脑中的表现与其他概念不同。它们更难学习。但一旦掌握，它们就会为你提供无与伦比的清晰和稳定的心理形象。这是通过数学真理和逻辑形式主义的独特性质实现的。

In your head, mathematical concepts behave differently than other concepts. They’re much more difficult to learn. But once in place, they provide you with mental images of an incomparable clarity and stability. This is made possible by the unique properties of mathematical truth and logical formalism.

你第一次了解大象是在小时候。然后你了解到大象有两种，你可以通过耳朵的大小来区分它们。现在你知道大象有三种不同的种类。谁知道明天会有多少种呢？

You were a child when you first learned about elephants. Then you learned that there are two different kinds and you could tell them apart by the size of their ears. Now you know that there are three distinct species of elephants. Who knows how many there’ll be tomorrow?

有了数字 2，您就永远不会遇到这样的问题。数学真理将数学概念联系在一起，形成一个独特、连贯且稳定的心理矩阵。可能很难解释数字 2 到底是什么，但您知道 2 + 2 = 4，而且它不会改变。

With the number 2, you’ll never run into such issues. Mathematical truth ties mathematical concepts together to form a mental matrix that is uniquely coherent and stable. It may be difficult to explain what the number 2 really is, but you know that 2 + 2 = 4 and that it’s not going to change.

你的数学直觉永远不会变得完美，但逻辑和数学真理使你能够不断地改进和重新调整它。

Your mathematical intuition will never become perfect, but logic and mathematical truth enable you to continually refine and recalibrate it.

即使你认为自己数学很差，你头脑中已经存在的数学所形成的概念矩阵是你与世界关系的最坚实的锚点。没有数字，没有圆形和正方形，没有你对三维空间中的点和轨迹的感知，没有x和y，没有距离、速度和加速度的概念，没有直线可以无限延伸的想法，没有概率，没有加法和乘法，没有真理和逻辑推理的概念，你周围的整个世界就会突然变得如此模糊和不稳定，让你感觉自己像是被做了脑白质切除术一样。

Even if you think you’re terrible at math, the conceptual matrix formed by the math that already lives inside your head is the most solid anchor point of your relationship to the world. Without numbers, without circles and squares, without your perception of points and trajectories in a three-dimensional space, without x and y, without the concepts of distance, speed, and acceleration, without the idea that a straight line can continue infinitely, without probabilities, without addition and multiplication, without the very notion of truth and logical reasoning, the whole world around you would suddenly become so blurred and unsteady that you’d feel like you’d been lobotomized.

你所理解的数学知识可以增强现实，并为你增添一层神奇的可理解性。它让你变得超级清醒。

The math that you understand augments reality and adds a magical layer of intelligibility. It makes you hyperlucid.

随着时间的推移，这些数学对你来说变得如此具体和明显，如此“真实”，以至于它不再感觉像数学。相比之下，你还不理解的数学总是显得抽象、荒谬和“虚构”。

With time, this math has become so concrete and obvious to you, so “real,” that it no longer feels like math. By comparison, the math that you don’t yet understand will always seem abstract, absurd, “imaginary.”

然而，这些看似显而易见、根深蒂固的概念并不总是存在的。很难相信，但像整数这样简单的东西需要人们通过思维的力量，在人类理解的范围内寻找它们。他们首先感觉到它们在直觉的迷雾中孵化。然后他们努力用语言表达它们。他们努力让这些文字简单易懂，这样每个人都能清楚地看到它们。

And yet these concepts that seem so evident and deeply embedded in you weren’t always in the picture. It’s difficult to believe, but things as simple as whole numbers required people to seek them out, through the power of thought, in the confines of human understanding. They first felt them hatching in the fog of their intuition. Then they struggled to put words to them. They worked to make these words simple and accessible, so that everyone could end up seeing them clearly.

当今数学家的头脑中蕴藏着比你们所学到的知识多一千倍的知识。

Inside the heads of mathematicians today, there’s a thousand times more than all you’ve been taught.

数学不是宇宙的语言。数学让我们能够清晰准确地表达那些我们无法用手指指点的东西。数学让我们能够推理和进行科学研究。数学让我们成为现在的样子，无论好坏。

Mathematics isn’t the language of the universe. It’s the language that allows us to speak with clarity and precision of all the things that we can’t point to with our fingers. It’s the language that makes us capable of reasoning and doing science. It’s the language that’s made us what we are, for better and for worse.

这种将数学视为一种心理重新编程和人类感知扩展的技术的方法相当新颖。这种观点已经存在了一段时间，但没有人花时间去澄清它并让公众能够理解它——至少直到最近。

This way of approaching math, as a technique of mental reprogramming and extension of human perception, is fairly recent. It’s a vision that had been in the air for some time without anyone taking the time to clarify it and make it accessible to the general public—at least until quite recently.

瑟写的这些引人注目的诗句很好地表达了这一点2011 年，斯通说：“人们普遍认为数学是对普遍真理的追求，是对不局限于任何单一固定背景的模式的追求。但从更深层次来看，数学的目标是开发出人类观察和思考世界的更好方式。数学是一段变革之旅，数学的进步可以通过我们思维方式的变化来更好地衡量，而不是通过我们发现的外部真理来衡量。”

It is beautifully expressed in these striking lines written by Thurston in 2011: “Mathematics is commonly thought to be the pursuit of universal truths, of patterns that are not anchored to any single fixed context. But on a deeper level the goal of mathematics is to develop enhanced ways for humans to see and think about the world. Mathematics is a transforming journey, and progress in it can be better measured by changes in how we think than by the external truths we discover.”

虚构作品

A Work of Fiction

仍有一个关键点，也许是本书最重要的一点。

There remains one crucial point, perhaps the most important of the book.

多年来困扰我的这种奇怪的感觉与数学有什么用处这个问题无关。

The weird feeling that had troubled me throughout the years had nothing to do with the question of what math is good for.

问这个问题的人不做数学。做数学的人很清楚数学有好处，哪怕只是给他们带来快乐，那种随着数学进步，世界变得越来越光明的神奇感觉。

People who ask that question don’t do math. People who do math know quite well it’s good for something, if only to give them pleasure, that magical feeling of seeing the world become more and more illuminated the further they progress in math.

这种奇怪的感觉与成为一名数学家的实际经历以及内心的感受更相关。这与我所准备的完全不同。确实发生了一些奇怪的事情。

The weird feeling had more to do with the actual experience of becoming a mathematician and what it felt like internally. It wasn’t like anything I was prepared for. Something really strange was going on.

我将解释它是什么，但在此之前，我需要首先对数学中最令人不安的方面做一些观察：不断提及那些“实际上”不存在的事物，而你无论如何都必须试着去想象。

I’m going to explain what it was, but before I do, I first need to make a few observations on what is quite possibly the most disconcerting aspect of math: the constant reference to things that don’t exist “for real” and that you have to try to imagine anyway.

对于那些想要了解数学的人，你可以给出的最简单、最基本的建议，也是我在这本书中反复提到的，就是假装事物真的存在，就在你面前，你可以伸手触摸到它们。

The most simple and fundamental advice you can give to people who want to understand math, which I’ve repeated throughout this book, is to pretend the things are really there, right in front of you, and that you can reach out and touch them.

不懂数学的人基本上处于怀疑状态。他们拒绝想象实际上不存在的事物，因为他们不明白其中的意义。这对他们来说毫无意义。

People who don’t understand math are basically stuck in a state of disbelief. They’re refusing to imagine things that don’t actually exist, because they don’t see the point. It just makes no sense to them.

我承认这令人不安，但赋予数学意义的唯一方法是想象它所谈论的东西确实存在。格罗滕迪克在这段话中对此非常坦诚，我之前曾引用过这段话：“我一生都无法阅读数学文本，无论它多么琐碎或简单，除非我能够根据我对数学事物的经验赋予它‘意义’，也就是说，除非文本在我心中唤起心理意象，即赋予它生命的直觉。”

I admit that it’s disconcerting, but the only way to give meaning to mathematics is to imagine that the things it’s talking about really exist. Grothendieck is very transparent about it in this passage, which I’ve previously cited: “All my life I’ve been unable to read a mathematical text, however trivial or simple it may be, unless I’m able to give this text a ‘meaning’ in terms of my experience of mathematical things, that is unless the text arouses in me mental images, intuitions that will give it life.”

正如我们所见，这些心理意象既不宏伟也不复杂。它们总是幼稚、简单，而且几乎总是错误的。当数学家思考球体时，他们想象球体的方式与你大致相同。

As we’ve seen, there’s nothing grandiose or sophisticated about these mental images. They’re always childish, always simplistic, and almost always plainly wrong. When mathematicians think about spheres, they imagine them more or less the same way you do.

数学家也是人，他们只能通过感知的方式理解数学对象，通过错误的人类解释、近似值、将数学词汇翻译成人类语言。

Mathematicians are human beings. They can understand mathematical objects only in a perceptual manner, via false human interpretations, approximations, translations from mathematical vocabulary into human language.

事实上，这正是数学对我们如此有益的原因：它迫使我们丰富我们的人类词汇和人类感知。

In fact, this is precisely why math is so beneficial for us: it forces us to enrich our human vocabulary and our human perception.

另一方面，数学家始终牢记，他们心中的图像只是对事实的近似，并且他们不断寻找找出他们图像的错误之处。

On the other hand, mathematicians always keep in mind that their mental pictures are only an approximation of the truth, and they’re constantly looking to find out how their pictures are false.

真正的球体存在于其他地方，在某种平行宇宙中。知道这个平行宇宙是否真的存在是一场无意义的争论，因为它无论如何都无法进入。一些数学家确信它存在，另一些人确信它不存在——还有一些人，比如我，对这两种说法都不在乎。

Real spheres exist elsewhere, in a sort of parallel universe. Knowing whether or not this parallel universe really exists is a useless debate, since it’s inaccessible anyway. Some mathematicians are convinced that it exists, others are convinced that it doesn’t—and still others, like me, couldn’t care either way.

唯一重要的事情（这也是真正令人不安的地方）是，你必须坚决地“假装”这个平行宇宙存在，因为如果你不这样做，数学只不过是一张纸上的一堆神秘符号而已。

The only thing that counts (and this is where it really becomes disconcerting) is that you must imperatively act “as if” this parallel universe existed, because if you don’t mathematics is nothing more than a bunch of cryptic symbols on a piece of paper.

这也解释了为什么数学家们坚持谈论数学对象来表示大多数人所说的数学抽象。

This explains the insistence of mathematicians on speaking of mathematical objects to designate what most people call mathematical abstractions.

换句话说，从纯粹实用的角度来看，数学与虚构没有区别。

In other words, from a purely practical standpoint, math is indistinguishable from fiction.

学习数学是一种纯粹的想象活动。我们通过思维的力量将数学对象带入我们的头脑，并通过一种神秘成分的凝聚作用将它们结合在一起，这种神秘成分在某种程度上是小说的真正英雄：数学真理。

Learning math is an activity of pure imagination. We bring mathematical objects into our heads through the power of thought and keep them together there through the cohesive effect of a mysterious ingredient, which in a way is the true hero of the fiction: mathematical truth.

在所有数学概念中，真理是最简单也是最难解释的。如果你想解释数字 2，你可以举起两个橘子。如果你想解释什么是三角形，你可以指向一个三角形。但是，对于数学家来说，你能举起或指向什么来解释什么是真理呢？

Of all the mathematical concepts, truth is at once the simplest and the most difficult to explain. If you want to explain the number 2, you can hold up two oranges. If you want to explain what a triangle is, you can point to a triangle. But what can you hold up or point to in order to explain what truth is for mathematicians?

不管你信不信，虚构的故事是有效的。数学家开发了新方法来研究现实，以及新的思维方式，这些方法在整个历史中都证明了其有效性。

Believe it or not, the fiction works. Mathematicians develop new ways of approaching reality and new ways of thinking that, throughout history, have demonstrated their effectiveness.

小说中的对象通过具体直观的化身，成为丰富我们对世界的理解的新概念。它们仿佛走出了小说，成为“现实”，化身为现实，就像我们看到两个橘子时，数字2就变成了现实。

The objects of the fiction, via their concrete and intuitive incarnation, become new concepts that enrich our understanding of the world. It’s as if they stepped out of the fiction to become “real,” incarnate, like the number 2 becomes real when we see two oranges.

在回归现实的过程中，所有数学对象都失去了其完美性，但它们保留了使它们成为虚构事物的本质特征。橙子可能不是真正的球体，但它仍然是圆的。

In the process of returning to reality, all mathematical objects lose their perfection, but they conserve the essential characteristics that made them what they were in the fiction. An orange may not be a real sphere, but it’s still round.

除一个之外，所有数学对象都存在。主角仍然停留在小说中，因为现实世界中没有任何东西与数学真理有任何相似之处。

All mathematical objects, that is, save one. The central character stays stuck in the fiction, as nothing in the real world even remotely resembles mathematical truth.

当梦想消逝的那一刻，数学真理也瞬间消失，就像精灵回到瓶子里一样。

The moment the dream fades away, mathematical truth instantly vanishes, like a genie going back into its bottle.

想象中的朋友

An Imaginary Friend

我还没有解释是什么导致了我的奇怪感觉，但很有可能你自己也开始体验到一种奇怪的感觉。

I haven’t yet explained what was causing my weird feeling, but it’s quite possible that you’re starting to experience a weird feeling of your own.

而且，说实话，确实有些奇怪。数学家们自以为的“理性”和他们实际做的事情不仅存在脱节，而且我们越深入挖掘，就越觉得奇怪。

And, to be honest, there is indeed something weird. Not only is there a disconnect between the assumed “rationality” of mathematicians and the strangeness of what they actually do, but the deeper we dig, the stranger it becomes.

我不知道还有什么人类活动会涉及如此激烈的现实与虚构之间的来回。从这个角度来看，这种方法似乎从一开始就完全是疯狂的，注定要失败。这有点像数学家在和一个想象中的朋友交谈，这个朋友会告诉他们周围世界的秘密。这怎么可能成功呢？

I am not aware of any other human activity that involves such a violent back-and-forth between reality and fiction. Seen in this light, the approach as a whole seems utterly insane and doomed from the start. It’s a bit like mathematicians were having conversations with an imaginary friend who lets them in on secrets about the world around them. How could this have the slightest chance of success?

这种内在的怪异性在整个数学历史中一直困扰着人们对数学的理解。

This intrinsic weirdness has clouded the understanding of mathematics throughout its history.

存在所谓的“虚数”，其虚数既不大于所谓的“实数”，也不小于所谓的“有理数”。

There are so-called “imaginary” numbers that are neither more nor less imaginary than so-called “real” numbers, which are neither more nor less real than so-called “rational” numbers.

每当一种新类型的数字被引入时，都会引起很多不安，不仅是公众，而且包括那些引入新数字的数学家本身。

Each time a new type of number was introduced, it provoked a lot of unease, not only among the public but also among mathematicians themselves, including those who had introduced the new numbers.

十九世纪，仍有严肃的数学家声称负数不过是童话故事。在十五、十六世纪，甚至连他们的拥护者也称负数为荒谬的数字。从那时起，现实似乎已经发生了变化，并决定改变立场。这些以前荒谬的数字已经变得具体而熟悉。它们已经占据了日常生活。要向自己证明负数不是童话故事，你只需开一个银行账户即可。

In the nineteenth century, there were still serious mathematicians who claimed that negative numbers were nothing but a fairy tale. In the fifteenth and sixteenth centuries even their advocates labeled them absurd numbers. Since then, it’s as if reality itself had changed, and decided to switch sides. These previously absurd numbers have become concrete and familiar. They’ve taken over everyday life. To prove to yourself that negative numbers aren’t fairy tales, you just have to open a bank account.

正如我们所看到的，康托尔因为冷静而准确地谈论了无限而被贴上了“科学骗子”、“叛徒”和“腐蚀青年”的标签。人们真正责备他的是，他把本应转瞬即逝的东西变成了有形的东西。从神学的角度来看，数学是不公平的竞争。

As we saw, Cantor was labeled a “scientific charlatan,” a “renegade,” a “corruptor of youth” for having talked about infinity calmly and precisely. What people really reproached him for was having made tangible what should have stayed evanescent. From a theological perspective, mathematics is unfair competition.

康托尔宣称：“数学的本质就是自由。”数学家的自由就是从“真实”的那一刻起，就将“虚构”的事物视为“真实”的事物。最终他们甚至认为它们是“显而易见的”。

“The essence of mathematics is its freedom,” declared Cantor. The freedom of mathematicians is to treat “imaginary” things as “real” things from the moment they are “true.” In the end they even see them as being “obvious.”

这种方法效果非常好。显然，数学家们不会在事情进展顺利时就此止步。他们继续用他们构造的超自然或神奇性质来取悦自己。他们操纵“理想”和“消失光谱”。著名的“怪物月光猜想”（以“怪物”命名，怪物是 196,883 维的物体）是用“无鬼定理”证明的。在代数中，有一种构造称为艾伦伯格骗局。

It happens that this approach works remarkably well. Mathematicians obviously aren’t going to stop when things are going so well. They continue to amuse themselves with the supernatural or miraculous nature of their constructions. They manipulate “ideals” and “vanishing spectra.” The famous “monstrous moonshine conjectures” (named after the “Monster,” an object that lives in dimension 196,883) were proved using a “no-ghost theorem.” In algebra, there’s a construction called the Eilenberg swindle.

如果理解数学的过程已经相当奇特，那么发现数学的过程就更奇特了。这种体验如此奇特和令人不安，以至于大多数描述看起来都像是神秘主义者写的。

If the process of understanding math is already quite bizarre, the discovery process is even more so. The experience is so singular and disconcerting that most accounts look like they were written by mystics.

最令人困惑的方面之一是想法突然出现，毫不费力，而且几乎总是不方便。正如格罗滕迪克所说，它们“仿佛从虚空中被召唤而来”。

One of the most baffling aspects is the abrupt manner in which ideas come to you, without effort and almost always inconveniently. They emerge, as Grothendieck puts it, “as if summoned from the void.”

在鲍勃·托马森和汤姆·特罗博撰写的一篇颇具影响力的研究文章中，我们得知第二位作者只是在他死后才做出贡献，他出现在第一位作者的梦中。他不仅提出了正确的方法，还阻止了第一位作者轻易地将其视为无望：“汤姆的模拟人如此坚持，我知道在我弄清楚论点之前，他不会让我安然入睡。”

In an influential research article by Bob Thomason and Tom Trobaugh, we’re told that the second author contributed only after he was dead, by means of appearing in a dream of the first author. Not only did he suggest the right approach, he stopped the first author from readily dismissing it as hopeless: “Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument.”

我的一位好友是一位出色的数学家（我不愿透露他的名字），最近他告诉我，他有一种强烈的印象（他从不敢与他人分享），那就是他职业生涯中最伟大的想法都是上帝直接启发的（尽管他是一位公开的无神论者）。

One of my close friends, an excellent mathematician whose name I won’t disclose, recently told me that he had the distinct impression (which he never dared share with others) that the greatest ideas in his career had been directly suggested by God (even though he’s an avowed atheist).

就我而言，我从来没有过类似的感觉。我只是觉得能够飘浮并穿过墙壁。

For my part, I’ve never felt anything along those lines. I’ve simply had the impression of being able to levitate and pass through walls.

严守秘密

Well-Guarded Secrets

这真是件奇怪的事情。我越进步，越深入数学的核心，越学会掌握促进深刻理解和创造力的技巧，它就越像巫术和黑魔法。

That was the weird thing. The more I advanced, the further I dove into the heart of mathematics, the more I learned to master the techniques that facilitate deep understanding and creativity, the more it began to resemble witchcraft and black magic.

笛卡尔认为数学家之所以保守秘密，是因为害怕失去声誉。他认为，如果人们知道有一种方法，而且方法如此简单，他们就不会再把数学家视为半神，而是会意识到他们只是普通人。

Descartes thought that mathematicians guarded their secrets for fear of losing their prestige. If people knew that there was a method and it was that simple, he reckoned, they would stop looking at mathematicians like they were demigods, and come to the realization that they’re just normal people.

真正的解释无疑更加简单：数学家只是害怕被称为疯子。

The real explanation is undoubtedly more trivial: mathematicians are simply afraid of being called insane.

如果我自己没有成为他们中的一员，我可能会继续相信他们是能够说宇宙语言的半神。但我知道这不是真的。我知道自己来自哪里。我知道是什么让我变得更好。每一个关键步骤总是或多或少偶然地发现一种克服我的抑制的新技术或一种让我的想象力发挥作用的新方法。

If I hadn’t become one of them myself, I might have continued to believe that they were demigods capable of speaking the language of the universe. But I know it’s not true. I know where I come from. I saw what made me get better. Each key step was always the more or less fortuitous discovery of a new technique to overcome my inhibitions or a new way of making my imagination work.

实际上，数学与自然科学没有太大关系。它与心理学更相关，是心理学的一种深奥且实用的分支。

In practice, mathematics doesn’t have much to do with the hard sciences. It’s rather more related to psychology, of which it’s a kind of esoteric and applied sub-branch.

不可否认，数学创造让人感觉神奇而超自然。但在这一切背后，必然存在着一种既不超自然也不神奇的人类现实。

It’s undeniable that mathematical creation feels magical and supernatural. But behind all that there’s necessarily a human reality that is neither supernatural nor magical.

真正困扰我并让我想继续探索这些主题直到我觉得能够以简单的方式讲述这个故事的是一种巨大的浪费感。

What really troubled me and made me want to continue to explore these subjects until I felt able to tell the story in a simple manner was an immense feeling of waste.

人类没有任何其他项目像数学一样享有如此的声望和知识权威。如果数学家无法解释他们的方法而不给人留下他们是某种巫师的印象，那并不意味着他们真的是巫师。

No other human project has the prestige and intellectual authority as that of mathematics. If mathematicians are incapable of explaining their approach without giving the impression that they’re some kind of shamans, that doesn’t mean they’re actual shamans.

这仅仅意味着他们没有使用正确的词语并且他们的解释不完整。

It simply means that they’re not using the right words and that their explanation is incomplete.

挥手的正确方式

The Right Way of Waving My Hands

为什么教数学这么难？为什么几个世纪以来一直如此？我们未能分享和交流什么？我们到底错过了什么？

Why is it so hard to teach math? Why has this been so for centuries and centuries? What do we fail to share and communicate? What exactly are we missing?

我学习数学是因为我无法理解如何理解数学。我期望有人能向我解释为什么这是可能的，以及如何做到这一点。解释从未出现。这个话题从未被提起过。

I studied math because I couldn’t understand how it was possible to understand it. I expected someone would explain to me why it was possible and how to do it. The explanation never came. The subject was never even raised.

这并没有阻止我自学。和许多人一样，我对自己最有价值的数学方面保持沉默，这让我很沮丧。

That didn’t stop me from learning on my own. Like so many others, I experienced the frustration of staying silent about the aspect of mathematics that, for me, was of the most value.

每当我发现自己处于教学或解释我的作品的境地时，我都试图将两个层次的论述结合在一起：由严格的定义和精确的陈述组成的正式层次，以及由正确的隐喻、正确的图画、正确的语调和正确的挥手方式组成的直观层次。

Each time I found myself in a situation of teaching or explaining my work, I tried to bring together two levels of discourse: a formal level made of rigorous definitions and precise statements, and an intuitive level, with the right metaphors, the right drawings, the right inflection of my voice, the right way of waving my hands.

这两个层次是相辅相成的。正式的讲座不分享自己的直觉，只表达自己的动机是毫无意义的。但是，没有任何形式化的纯粹直觉的论述同样毫无意义——这就是为什么普及数学的尝试常常会失败。一旦你摆脱了官方数学，直觉就会失去它的立足点。

These two levels complement one another. A formal lecture without motivation and without sharing your intuition is meaningless. But a purely intuitive discourse without any formalization is equally meaningless—this is why attempts at popularizing math so often miss the mark. Once you get rid of official math, intuition loses its moorings.

如果没有正式定义和正式陈述，数学的教学量就会受到限制。到了一定程度，没有了形式主义，数学就不再是数学，而只是人们挥挥手而已。

There is an inherent limit to how much math can be taught without formal definitions and formal statements. At some point, without formalism, it’s no longer mathematics, it’s just people waving their hands.

在我离开学术界前不久，我有机会讲授职业生涯中最有趣的一门课程。这是一门为期一学期的数学入门课程，面向法国最负盛名的高等师范学院的文学和哲学系学生。

Shortly before quitting academia, I had the opportunity to give the most interesting course of my career. It was a semester-long introductory math course for literature and philosophy students at the école normale supérieure, one of the most prestigious universities in France.

这是一个进行实验的机会，让我可以面对这个基本问题：你能教授在脑海中思考数学的艺术吗？

It was an opportunity to experiment, and confront myself with this fundamental question: can you teach the art of seeing math in your head?

我重新回到了传统上所谓的数学基础：逻辑和集合论。那时我才意识到自己一直走错了方向。逻辑和集合论不是数学的基础，而是数学的分支。它们的重点是证明概念的数学形式化——这是一个完全合法的研究领域，但它并不能真正解释数学是什么，更不用说如何教授数学了。

I plunged back into what are traditionally called the foundations of mathematics: logic and set theory. That’s when I realized that I had been going about it the wrong way. Logic and set theory are not foundations of mathematics, they are branches of mathematics. Their focus is the mathematical formalization of the notion of proof—a perfectly legitimate field of study, but one that won’t bring much clarity to what math really is, let alone how to teach it.

本书中的某些观点和例子直接源自我当时的课堂笔记。然而，当时我缺少一个关键要素。

Certain ideas and examples in this book go back directly to my class notes from this time. Back then, however, I was missing a crucial ingredient.

在课堂上，我总觉得有些不对劲，好像我没有把谈话放在正确的位置。我热爱我内心深处的数学，但我无法用别人能理解的语言来解释它。

In my class, I constantly felt that something wasn’t right, as if I hadn’t managed to situate the conversation in the right place. I loved the math that was alive inside of me, but I was unable to explain it in words that others could relate to.

正是在这种背景下，我宣布结束我的科学生涯。做出这样的决定从来都不容易。试图找出一个可以解释这一切的因素是幼稚的。然而，在众多因素中，有一个特别令人沮丧的地方：我无法以有意义的方式教授数学。我可以教授数学应该是什么，但我无法教授它对我来说真正是什么。不知何故，感觉好像这种真诚的教学水平是不被允许的，好像一个古老的禁忌阻止了它的发生。

It was in this context that I called an end to my scientific career. Decisions of this kind are never easy to make. Trying to pinpoint a single factor that explains it all would be naïve. Among the multitude of factors, there was, however, this particular frustration: I wasn’t able to teach math in a meaningful way. I could teach what math was supposed to be, but I couldn’t teach what it really was for me. Somehow, it felt as if this sincere level of teaching wasn’t permitted, as if an ancient taboo was preventing it from happening.

现在回想起来，很明显我的入门课程缺少对人类理解数学的经验的讨论。

With hindsight, it’s now clear that my introductory course was simply missing a discussion of the human experience of understanding mathematics.

不可能的故事

The Impossible Story

欧几里得的《几何原本》是历史上最具影响力的数学专著。它的历史可以追溯到两千三百年前，几个世纪以来，它已经成为数学推理本身的蓝图。

Euclid’s Elements is the most influential mathematics treatise in history. It dates back twenty-three hundred years and, throughout the centuries, it has come to serve as a blueprint for mathematical reasoning itself.

从那时起，数学就被描述为逻辑推理的科学。故事的另一部分，即我们在头脑中进行的看不见的行为，一直被掩盖着。这当然不是阴谋。一个更合理的解释是，在两千三百年里，故事的另一部分根本就不可能被讲述。

Since this time, mathematics has been presented as the science of logical deduction. The other part of the story, that which concerns the unseen actions that we perform in our heads, has been obscured. This is of course no conspiracy. A more plausible explanation is that, for twenty-three hundred years, this other part of the story was simply impossible to tell.

讲述这个故事需要解释我们脑子里在想什么，而我们没有令人满意的方式来表达我们自身智力的内部运作。

Telling the story would have required explaining what was going on in our heads, and we had no satisfying way of representing the inner workings of our own intelligence.

我们能想到的唯一模型是机械演绎推理，符合欧几里得《几何原本》的精神。将智力视为计算能力的历史与数学本身一样悠久。“计算”来自拉丁语calculus，意思是“小卵石”，指算盘上用来计数的石头。几个世纪以来，人们把大脑比作算盘，然后是齿轮计算机，然后是硅片。这个比喻将机械演绎推理与数学和理性——以及智力本身——混为一谈。

The only model we could think of was that of mechanical deductive reasoning, in the spirit of Euclid’s Elements. Viewing intelligence as the ability to perform calculations is as old as math itself. “Calculation” comes from the Latin calculus, which means “small pebble,” referring to the stones used on an abacus for counting. Across the centuries, the brain was compared to an abacus, then a geared computing machine, then a silicon chip. The metaphor conflated mechanical deductive reasoning with mathematics and rationality—and with intelligence itself.

正如我们所见，这个比喻存在严重缺陷。这种对大脑过程的严重误解使我们无法将数学与人类的共同经验联系起来。

As we’ve seen, this metaphor is deeply flawed. This profound misinterpretation of our brain processes has made us unable to relate mathematics to the common human experience.

讽刺的是，我们一直都知道，我们的智力不能被简化为计算。我们内心深处知道有其他事情正在发生。然而，如果不求助于超自然力量，我们就无法唤起它。我们只能依靠灵魂和直觉、第三只眼睛和第六感。我们想象着魔法实体在我们控制之外行动，只有少数拥有特殊天赋的精英才有特权直接与它们交流。这些模型自史前时代以来几乎没有改变过。

Ironically, we’ve always known that our intelligence couldn’t be reduced to calculations. We’ve known deep inside that something else was going on. Yet we had no way to evoke it without having recourse to the supernatural. We were stuck with spirits and intuitions, third eyes and sixth senses. We imagined magical entities acting outside of our control, and with which only a small elite, endowed with a special gift, had the privilege of being able to directly communicate. These models had remained practically unchanged since prehistory.

我们的语言本身就是一个谜。谁发明了文字？什么是概念？什么是意义？什么是真理？我们怎么才能理解短语？几千年来，这些甚至都不是科学问题——它们属于形而上学和神学领域。

Our language itself was a mystery. Who invented words? What is a concept? What is meaning? What is truth? How are we able to even make sense of phrases? For millennia, these weren’t even science questions—they belonged to the fields of metaphysics and theology.

理解数学就是重新编程你的直觉。首先，这是神经可塑性的问题。数学家的秘密技巧与本·安德伍德通过咂舌看世界的技巧一样超自然。

To understand math is to reprogram your intuition. It is, above all, a matter of neuroplasticity. The secret techniques of mathematicians are neither more nor less paranormal than those that allowed Ben Underwood to see the world by clicking his tongue.

如果我们把我们的心理活动视为某种神奇的东西，那么数学从根本上来说就无法解释。但随着我们进入人工智能时代，我们可能终于迎来了突破口。

As long as we treated our mental activity as something magical, mathematics was fundamentally impossible to explain. But as we’re entering the age of artificial intelligence, we may finally have an opening.

正是我与深度学习算法的邂逅让我得以撰写这本书。这是我一生中第一次接触到大脑隐喻，让我能够理解自己的旅程。我意识到我的证言很有价值，因为我可以描述我的主题以足够清晰、足够“理性”的术语来表达经验，可以摆脱私人谈话的安全范围。

It was my encounter with deep-learning algorithms that enabled me to write this book. For the first time in my life, I had access to a brain metaphor that allowed me to make sense of my own journey. I realized that my testimonial had value, as I could describe my subjective experience in terms that were sufficiently clear, sufficiently “rational,” to escape the safe confines of private conversations.

从此我不再将我的故事视为一个不可能的故事。

This is how I ceased to view my story as an impossible story.

当我用深度学习的比喻来看待它们时，困扰我多年的奇怪现象突然变得不再那么奇怪了。是的，想法会出乎意料地出现，“就像从虚空中召唤而来”，但这是正常的。是的，可塑性是一种缓慢而无声的机​​制，只要我们接触到正确的心理意象，它就会发生，我们不需要付出任何真正的努力。是的，我们正是在强迫自己想象我们还不理解的事物时才学到东西，不幸的是，这正是大多数人逃避的事情。是的，关注困扰我们的小细节至关重要，最快的学习方法是遵循最大困惑的道路。在这个框架内，笛卡尔怀疑可以被解释为一种加速我们学习的“对抗性”黑客攻击。

When I look at them with the deep-learning metaphor in mind, the strange phenomena that have troubled me for so many years suddenly cease being strange to me. Yes, ideas come unexpectedly, “as if summoned from the void,” but that’s normal. Yes, plasticity is a slow and silent mechanism that occurs without any real effort on our part, provided we’re exposed to the right mental images. Yes, we learn precisely when we force ourselves to imagine things that we don’t yet understand, which unfortunately is the same exact thing that most people run away from. Yes, paying attention to the small details that trouble us is of the utmost importance, and the fastest way to learn is to follow the path of maximum perplexity. Cartesian doubt, within this framework, can be interpreted as an “adversarial” hack to accelerate our learning.

数学的觉醒

A Mathematical Awakening

几千年来，我们用一种大多数人无法理解的方式来表达数学。现在我们终于有机会用不同的方式谈论它了。

For millennia, we presented mathematics in a way that made it unintelligible to most people. We finally have the opportunity to talk about it differently.

学习数学应该像学习其他运动技能一样，比如学习游泳或骑自行车，每个人都应该能够学习。我们对语言本质和思维功能的错误信念阻碍了这种简单而直接的学习。它们灌输恐惧和抑制，阻碍了看不见的行动，没有这些行动，数学学习就不可能进行。

Learning math should be like learning any other motor skill, like learning to swim or ride a bike, and it should be accessible to everyone. Our false beliefs about the nature of our language and the functioning of our thought are obstacles to this simple and direct learning. They instill fears and inhibitions that block the unseen actions without which no mathematical learning can take place.

你如何向那些相信自己的直觉和对现实的感知是天生的、不可能被重新编程的人教授数学？这就像向那些相信他们的身体就像岩石一样坚硬，会下沉。任何成功的教导的前提都是要摆脱这样的信念。

How do you teach math to someone who believes that their intuition and perception of reality are given and impossible to reprogram? It’s exactly like teaching swimming to someone who is convinced their body is as dense as rock and will sink. A prelude to any successful teaching is getting rid of such beliefs.

这就是为什么我将这本书设想为一本关于觉醒和解放的书。我相信数学教会了我关于我们的身体、它们如何运作以及我们可以用它们完成什么的重要课程，我想分享这些课程。

This is why I conceived of this book as a book of awakening and emancipation. I believe that mathematics has taught me important lessons about our bodies, how they function, and what we can accomplish with them, and I wanted to share those lessons.

我试图尽可能坦率地谈论我们头脑中的想法：主观现实、情感历程、身体和感官体验、我们所做事情的实际方面、我们如何做、它如何运作以及感觉如何。所有这些都不是教学的一部分，因为它不应该是数学的一部分，也因为我们对任何主观的东西都根深蒂固的怀疑。

I’ve tried to speak as candidly as possible about what goes on inside our heads: the subjective reality, the emotional journey, the physical and sensorial experience, the practical aspects of what we do, how we do it, how it works, and what it feels like. None of that has ever been part of teaching, because it wasn’t supposed to be part of math, and also because of our deeply ingrained suspicion toward anything subjective.

我很清楚主观性的缺点。笛卡尔写道：“我知道我们很可能因为自己的原因而犯错”，而这种情况自他那个时代以来就没有改善过。

I’m well aware of the shortcomings of subjectivity. “I know how likely we are to be wrong on our own account,” wrote Descartes, and the situation hasn’t improved since his time.

但我们真的有选择吗？当数学只是证明定理时，将主观体验视为二等公民是完全自然的，只有在时间允许的情况下才会非正式和轶事地涉及。但一旦我们意识到数学最终是关于人类理解的，这不再是一种选择。理解本质上是一种主观体验。

But do we really have a choice? When mathematics was just about proving theorems, it was entirely natural to treat the subjective experience as a second-rate citizen, to be covered only informally and anecdotally, if time allowed. But as soon as we realize that mathematics is ultimately about human understanding, this ceases to be an option. Understanding is, in essence, a subjective experience.

如果我的个人故事仅仅是“一个有天赋的人”的故事，那么它就没有什么价值。虽然我无法证明我没有特殊才能，但我知道，事实上，我自己对“有创造力的数学家必须在生物学上与众不同”这一观念极为恐惧，如果我找不到摆脱它的方法，我就会放弃。

My personal story would be of little value if it was simply that of “one with a gift.” While I cannot prove that I have no special talent, I know for a fact that I was myself extremely intimidated by the notion that creative mathematicians had to be biologically different, and would have given up if I hadn’t found a way to get rid of it.

从表面上看，数学创造力确实需要某种超人的智慧。要想摆脱“天赋”和“才能”的思维，就必须找到另一种解释。

From the outside, it indeed seems that mathematical creativity requires some sort of superhuman intelligence. To stop thinking in terms of “gifts” and “talents,” one has to find an alternate explanation.

我看待事物的方式，在我的职业生涯中，我最想做的就是想象富有创造力的数学家是黑客，他们找到了解锁我们认知“隐藏模式”的方法。大多数时候，他们是在不知不觉中做到的，而且完全无法解释他们是如何做到的。

My way of looking at things, which has served me well throughout my career, was to imagine that creative mathematicians were hackers who had found ways to unlock “hidden modes” of our cognition. Most of the time, they’d done so unwittingly, and were entirely incapable of explaining how.

这也是我写这本书时一直遵循的假设。我集中精力研究我亲身了解的少数事情，因此只研究了该主题的一小部分。

It’s also the hypothesis that’s guided me throughout the writing of this book. I concentrated on the few things that I knew firsthand, and thus on a small part of the subject.

我很幸运能够依靠笛卡尔、格罗滕迪克和瑟斯顿的著作。他们的故事彼此相似，就好像是从三个不同的角度讲述的同一个故事。这些故事与我亲身经历的相吻合，这让我能够将我的个人旅程铭刻在一个更古老、更强大、记录更丰富的传统中。

I was fortunate enough to be able to rely on the writings of Descartes, Grothendieck, and Thurston. Their stories closely resemble one another, as if it were the same story told from three different points of view. These stories are compatible with what I’ve lived through myself, which allowed me to inscribe my personal journey within a tradition that’s older, stronger, and more richly documented.

即便是他们，也没有掌握所有的关键。笛卡尔无法解释发生在他身上的事情，除非他做出二元论假设，即他的心灵具有神圣和非物质的本质，与他的身体分离。同样，格罗滕迪克也相信上帝在他耳边低语，在他的头脑里做梦。瑟斯顿是三人中最务实、最现代、最清醒的。

Even they didn’t have all the keys. Descartes was unable to account for what happened to him without making the dualist assumption that his mind was of a divine and immaterial essence, detached from his body. Similarly, Grothendieck was convinced that God whispered in his ear and dreamed inside his head. Thurston is the most pragmatic of the three, the most modern, and without doubt the most lucid.

他们的坦诚和对细节的关注是最重要的。他们尽力分享他们的经历以及他们认为的成功原因。

Their candor and sense of detail are of utmost value. They tried their best to share what they’d lived through and what, in their view, accounted for their success.

从第一页到最后一页，他们几乎只谈论想象力。每个人都描述了在新模式下如何运用想象力，这些新模式是偶然发现的，并且打破了他们所学的内容。

From the first page to the last, they speak almost only of imagination. Each describes the use of imagination in new modalities, discovered by accident and breaking with what they’d been taught.

格罗滕迪克将他作品的独特性归因于他对禁忌的超越：“似乎在所有自然科学中，只有在数学中，我所说的‘梦’或‘白日梦’才遭遇了明显绝对的禁令，这种禁令已有两千年以上的历史了。”

Grothendieck attributes the singularity of his work to his transgression of a taboo: “It would seem that among all the natural sciences, it is only in mathematics that what I call ‘the dream’ or ‘the daydream’ is struck with an apparently absolute interdiction, more than two millennia old.”

瑟斯顿的说法虽然不那么夸张，但同样具有影响力：“我认为，做白日梦不是一个缺陷，而是一个特性。”

Thurston puts it in a less grandiose but equally impactful fashion: “I have decided that daydreaming is not a bug but a feature.”

笛卡尔本人也是一位狂热的幻想家，他留给我们关于他强大技巧的引人入胜的描述。但讽刺的是，他也对有缺陷的理论负有责任，这使得我们很难认真对待他。

Descartes, himself an avid daydreamer, left us with a compelling account of his powerful techniques. But, by a cruel irony, he’s also responsible for the flawed theory that is making it so hard for us to take him seriously.

想象力是整个故事的关键。数学家们找到了一种独特的方式来运用想象力，并因此取得了巨大的成功。数学想象力更具远见和无限性，因为它受数学真理的引导，数学真理是让我们能够找出正确的想象事物的秘密成分，这些事物最终将巩固和扩展我们对周围世界的直觉理解。数学的真正基础就在这里，而不是形式逻辑或神学中。

Imagination is the key to this entire story. Mathematicians have found a unique way to use theirs and it has made them incredibly successful. Mathematical imagination is all the more visionary and limitless in that it is guided by mathematical truth, the secret ingredient that makes it possible to figure out the right things to imagine, the ones that will eventually solidify and expand our intuitive understanding of the world around us. This is where the true foundations of mathematics are to be found, and not in formal logic or theology.

但在狭隘的理性主义看来，这一切都没有任何意义。

But under a narrow-minded version of rationalism, none of this can make any sense.

我们被教导说，我们的思想对现实没有影响。对于一个受过教育的人来说，要肯定相反的观点需要很大的勇气。我们到处读到，人们可以通过思想的力量改变世界，但我们被训练成将其视为神秘主义或自我救助的虚张声势。

We’ve been taught that our thoughts have no impact on reality. For an educated person, it takes a lot of guts to affirm otherwise. We read here and there that one can change the world by the power of thought, but we’re trained to dismiss it as mysticism or self-help bluster.

这是二元论的精髓。我们的思想被认为是虚无缥缈的。你可以随心所欲地想象任何事物，而这对物质世界没有任何影响。

This is dualism at its finest. Our thoughts are presumed to be ethereal. You can imagine whatever you wish, however you wish, without that having any impact whatsoever on the physical world.

然而，运用想象力并不是在宇宙的虚空中航行。它也不是一种我们应该试图抑制的寄生活动。相反，它是一种真正的身体活动，是人类认知的核心。

Yet using our imagination isn’t navigating an ethereal layer of the cosmos. Nor is it a parasitical activity that we should seek to suppress. It is, instead, a genuine physical activity that is central to human cognition.

我们在头脑中看到和做的事情对我们神经学习的贡献与我们在现实中看到和做的事情一样大。如果我们在头脑中感到有冲动去执行看不见的动作，如果我们做梦，如果我们白天梦境，是因为它让我们能够虚构理解。我们的想象会改变我们大脑的实际线路，并真正改变我们看待世界的方式。

What we see and do in our heads contributes to neural learning every bit as much as what we see and do for real. If we feel the urge to perform unseen actions in our heads, if we dream and if we daydream, it’s because this allows us to fabricate understanding. What we imagine modifies the actual wiring of our brain and literally changes the way we see the world.

想象的方式有千百种。我们还没有学会认识它们，更不用说给它们命名了。思考、冥想、反思、想象、分析、幻想、推理、梦想：我们随意使用这些词，并不真正知道它们的含义，也没有意识到它们有多少共同之处。

There are a thousand and one ways to imagine. We haven’t yet learned to recognize them all, and still less to name them. Think, meditate, reflect, visualize, analyze, fantasize, reason, dream: we use these words haphazardly, without really knowing what they mean, and without realizing how much they have in common.

正是由于这种模糊性，所有的误解才会产生。甚至没有人愿意告诉我们，运用想象力有正确和错误的方法。有些方法会让我们变得愚蠢。有些方法会让我们发疯。而有些方法却能让我们变得异常聪明。

It’s through this vagueness that all the misunderstandings slip in. No one cared to even tell us that there are right and wrong ways of using our imagination. Some make us stupid. Some make us crazy. And some have the power of making us incredibly smart.

既然我们正在向机器传授智能的秘密，那么现在是时候开始向人类传授智能的秘密了。

Now that we’re teaching machines the secrets of intelligence, it’s about time we start teaching humans.